THESE DE DOCTORTA DE MATHEMATIQUES préparée à l'Institut ...fribergh/these.pdf · Alexander FRIBERGH Marches aléatoires en milieux aléatoires et phénomènes de ralentissement

THESE DE DOCTORAT DE MATHEMATIQUESDE L'UNIVERSITE CLAUDE BERNARD (LYON 1)préparée à l'Institut Camille JordanLaboratoire des MathématiquesUMR 5208 CNRS-UCBL

Thèse de doctorat

Specialité Mathématiques

presentée par

Alexander FRIBERGH

Marches aléatoires en milieuxaléatoires et phénomènes de

ralentissement

Rapporteurs :Noam BERGER Hebrew University of JerusalemPierre MATHIEU Université de Provence

Soutenue le 3 Juin 2009 devant le jury composé de :

Stéphane ATTAL Université Lyon1 ExaminateurVincent BEFFARA E.N.S. Lyon ExaminateurGérard BEN AROUS New-York University ExaminateurNina GANTERT Münster Universität ExaminateurYueyun HU Université Paris 13 ExaminateurPierre MATHIEU Université de Provence RapporteurChristophe SABOT Université Lyon 1 Directeur de thèse

ii

Remerciements

Je tiens à exprimer ma profonde reconnaissance à mon directeur de thèse, ChristopheSabot. Je tiens tout particulièrement à le remercier d'avoir ravivé mon intérêt pour lesmathématiques au travers de mon stage de master et de cette thèse. Durant les momentsdiciles de ces dernières années, ses idées, ses conseils avisés et son optimisme m'auronttoujours permis de me remettre à l'ouvrage plein d'espoir. Nos discussions ont été unélément central dans ma formation de chercheur, avoir pu apprendre à ses cotés est unhonneur.

Je suis extrêmement reconnaissant envers les diérentes personnes et institutions quim'ont accueilli durant cette thèse. En particulier, Nina Gantert à la Münster Universität,Serguei Popov à l'Universidade de São Paulo. Je suis également reconnaissant enversGérard Ben Arous et Alan Hammond du Courant Institute (New-York) avec qui j'aieu l'honneur de travailler. Finalement je remercie également l'Institut Mittag-Leer oùj'aurai eu le plaisir de passer plusieurs mois. Cette thèse doit énormément à toutes cesrencontres et ces voyages.

Je suis également très reconnaissant envers Noam Berger et Pierre Mathieu d'avoiraccepté d'être rapporteurs de ma thèse, malgré la tâche que cela représente. Lors denos rencontres, ils m'ont fait part de leur intéret pour mon travail ce qui m'a fait leplus grand plaisir. Je remercie également Gérard Ben Arous, Nina Gantert, Yueyun Hu,Vincent Beara et Stéphane Attal qui me font l'honneur de faire partie de mon jury.

Je tiens également à saluer Vincent Beara pour sa disponibilité et son enthousiasme.Je suis certain que nos discussions m'ont inspiré à plusieurs reprises.

Je remercie toutes les équipes administratives et techniques de l'Université Lyon 1et de L'École Normale Supérieure de Lyon pour leur gentillesse et leur ecacité.

Merci également à tous mes amis que j'ai pu cotoyer ces dernières années : Antoine,Damien, Eric, Frédéric, Gaël, Ion, Jean, Laurent, Maxime, Nicolas, Pierre ... j'en gardeun souvenir exceptionnel.

Finalement je voulais remercier ma famille qui a toujours cru en moi.

iii

Table des matières

1 Introduction 11 Origines du modèle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Formulation mathématique . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Organisation de la thèse . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Quelques modèles d'importance 71 La marche aléatoire en environnement aléatoire uni-dimensionnelle . . . . 7

1.1 Le modèle et son histoire . . . . . . . . . . . . . . . . . . . . . . . 71.2 Transience-récurrence et loi des grandes nombres . . . . . . . . . 81.3 Le cas récurrent : le potentiel de Sinaï . . . . . . . . . . . . . . . 91.4 Le cas transient à vitesse nulle . . . . . . . . . . . . . . . . . . . . 111.5 Principes de grandes déviations . . . . . . . . . . . . . . . . . . . 12

2 Marches aléatoires en milieu aléatoire sur des arbres . . . . . . . . . . . . 122.1 Modèle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2 Transience-récurrence . . . . . . . . . . . . . . . . . . . . . . . . . 142.3 Loi des grands nombres . . . . . . . . . . . . . . . . . . . . . . . . 142.4 Principes de grandes déviations . . . . . . . . . . . . . . . . . . . 15

3 Marches aléatoires en environnements aléatoires sur Zd avec d ≥ 2 . . . . 163.1 Le modèle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.2 Transience-récurrence . . . . . . . . . . . . . . . . . . . . . . . . . 173.3 Existence et étude de la vitesse . . . . . . . . . . . . . . . . . . . 183.4 Autres résultats . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4 Marches aléatoires sur des clusters de percolation . . . . . . . . . . . . . 204.1 La percolation par arêtes . . . . . . . . . . . . . . . . . . . . . . . 204.2 La marche aléatoire simple . . . . . . . . . . . . . . . . . . . . . . 214.3 La marche aléatoire biaisée . . . . . . . . . . . . . . . . . . . . . . 22

3 Présentation des résultats 251 Comportement de la vitesse sur le cluster de percolation vis-à-vis des

paramètres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

v

TABLE DES MATIÈRES

2 Un lien entre les M.A.M.A. et un modèle de piège jouet . . . . . . . . . . 293 Déviations modérées pour la M.A.M.A. sur Z . . . . . . . . . . . . . . . 36

4 Biased random walks on Galton-Watson trees with leaves 391 Introduction and statement of the results . . . . . . . . . . . . . . . . . . 402 Constructing the environment and the walk in the appropriate way . . . 443 Constructing a trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474 Sketch of the proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535 The time is essentially spent in big traps . . . . . . . . . . . . . . . . . . 546 Number of visits to a big trap . . . . . . . . . . . . . . . . . . . . . . . . 577 The time spent in dierent traps is asymptotically independent . . . . . 628 The time is spent at the bottom of the traps . . . . . . . . . . . . . . . . 659 Analysis of the time spent in big traps . . . . . . . . . . . . . . . . . . . 7010 Sums of i.i.d. random variables . . . . . . . . . . . . . . . . . . . . . . . 80

10.1 Computation of the Lévy spectral function . . . . . . . . . . . . . 8210.2 Computation of dλ . . . . . . . . . . . . . . . . . . . . . . . . . . 8410.3 Computation of the variance . . . . . . . . . . . . . . . . . . . . . 85

11 Limit theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8611.1 Proof of Theorem 1.3 . . . . . . . . . . . . . . . . . . . . . . . . 8611.2 Proof of Theorem 1.2 . . . . . . . . . . . . . . . . . . . . . . . . 8711.3 Proof of Theorem 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . 8911.4 Proof of Theorem 1.4 . . . . . . . . . . . . . . . . . . . . . . . . 93

5 The speed of a biased random walk on a percolation cluster at highdensity 971 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 972 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 983 Kalikow's auxiliary random walk . . . . . . . . . . . . . . . . . . . . . . 1034 Resistance estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1065 Percolation estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1156 Continuity of the speed at high density . . . . . . . . . . . . . . . . . . . 1237 Derivative of the speed at high density . . . . . . . . . . . . . . . . . . . 128

7.1 Another perturbed environment of Kalikow . . . . . . . . . . . . . 1287.2 Expansion of Green functions . . . . . . . . . . . . . . . . . . . . 1307.3 First order expansion of the asymptotic speed . . . . . . . . . . . 135

8 Estimate on Kalikow's environment . . . . . . . . . . . . . . . . . . . . . 1398.1 The perturbed hitting probabilites . . . . . . . . . . . . . . . . . 1408.2 Quenched estimates on perturbed Green functions . . . . . . . . . 1438.3 Decorrelation part . . . . . . . . . . . . . . . . . . . . . . . . . . 146

9 An increasing speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

vi

TABLE DES MATIÈRES

6 Slowdown and speedup of transient RWRE 1551 Introduction and results . . . . . . . . . . . . . . . . . . . . . . . . . . . 1562 More notations and some basic facts . . . . . . . . . . . . . . . . . . . . 1613 Estimates on the environment . . . . . . . . . . . . . . . . . . . . . . . . 1634 Bounds on the probability of connement . . . . . . . . . . . . . . . . . . 1695 Induced random walk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1736 Quenched slowdown . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

6.1 Time spent in a valley . . . . . . . . . . . . . . . . . . . . . . . . 1796.2 Time spent for backtracking . . . . . . . . . . . . . . . . . . . . . 1816.3 Time spent for the direct crossing . . . . . . . . . . . . . . . . . . 1826.4 Upper bound for the probability of quenched slowdown for the

hitting time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1836.5 Upper bound for the probability of quenched slowdown for the walk1856.6 Lower bound for quenched slowdown . . . . . . . . . . . . . . . . 186

7 Annealed slowdown . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1887.1 Lower bound for annealed slowdown . . . . . . . . . . . . . . . . 1887.2 Upper bound for annealed slowdown . . . . . . . . . . . . . . . . 190

8 Backtracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1918.1 Quenched backtracking for the hitting time . . . . . . . . . . . . . 1918.2 Quenched backtracking for the position of the random walk . . . . 1928.3 Annealed backtracking . . . . . . . . . . . . . . . . . . . . . . . . 194

9 Speedup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1949.1 Lower bound for the quenched probability of speedup . . . . . . . 1959.2 Upper bound for the quenched probability of speedup . . . . . . . 1979.3 Annealed speedup . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

vii

TABLE DES MATIÈRES

viii

1Introduction

1 Origines du modèle

L'intérêt porté par les mathématiciens aux marches aléatoires est probablement liéà la simplicité du modèle qui permet cependant de modéliser des phénomènes naturelscomplexes et donne lieu à des problèmes mathématiques diciles. L'un des modèles lesplus simples que l'on puisse considérer, comme son nom l'indique, est la marche aléatoiresimple sur Zd. On cherche à décrire le comportement d'un marcheur partant de l'originedu réseau de dimension d et qui saute à chaque unité de temps discrète vers l'un de ses2d voisins, ce dernier étant choisit uniformémement i.e. avec probabilité 1/(2d). Le casle plus simple, celui de la dimension d = 1, décrit l'évolution de la fortune d'un joueurdans un jeu de pile ou face.

Des quantités de questions naturelles se posent rapidement, on ne retiendra pourl'instant que deux.

Combien de fois le marcheur reviendra-t-il à l'origine ? Typiquement après un temps n où se situe le marcheur ?La première question remonte à Polya qui la considéra en 1921. La légende veut que

Polya rééchissait à ce probleme en marchant dans un parc près de Zürich alors qu'ilrencontrait constamment un couple de promeneurs. Informellement on peut résumerle théorème qu'il démontra sous la forme suivante : en dimension d ≤ 2 on revient

1

CHAPITRE 1. INTRODUCTION

inniment souvent à l'origine, on dit alors que la marche est récurrente, alors qu'endimension d ≥ 3 on revient un nombre ni (aléatoire) de fois à l'origine auquel cas elleest dite transiente. On résume souvent ce résultat en disant qu'un homme ivre nira parrentrer chez lui alors qu'un poisson ivre peut se perdre à jamais.

La deuxième question est plus délicate à formuler mathématiquement. Cependant ilest possible de dire qu'en un certain sens le marcheur se trouve à une distance

√n de

l'origine. De plus si on le regarde de plus en plus loin, son comportement ressemble àcelui d'un mouvement brownien. Cet objet tient son nom du biologiste Robert Brown quien 1828 observa que des grains de pollen suspendus dans l'eau eectuaient un mouvementcontinu et désordonné. Il fut énormément étudié à partir du vingtième siècle, car il estomniprésent dans des domaines aussi diérents que la nance (voir Bachelier [6]), laphysique (citons Einstein [32]) et bien sûr les mathématiques.

Evidemment ce modèle a ses limites pour décrire à lui seul des problèmes plus com-plexes, en eet il ne permet pas de décrire un mouvement dans un milieu hétérogène ouinconnu. Il faut donc être capable de modéliser et tenir compte de l'aléatoire induit parun environnement. Par exemple, dans une portion de désert essentiellement plate, nousallons rencontrer des imperfections, ce sont des dunes plus ou moins grandes qui vontinuencer notre marcheur, ce dernier étant plus enclin à contourner un tel obstacle.

C'est dans le but de pouvoir décrire ce genre de phénomènes que nous étudionsles marches aléatoires en milieux aléatoires. Nous utiliserons l'abréviation classique deM.A.M.A. pour désigner marches aléatoires en milieux aléatoires.

2 Formulation mathématique

Nous ne cherchons pas ici à donner une formulation générale des modèles de M.A.M.A.,i.e. sur des graphes généraux, on se contentera ici de formuler le problème sur le réseauZd pour les marches à plus proches voisins.

On note S le simplexe 2d-dimensionnel, posons Ω = SZd et notons la coordonnéede ω ∈ Ω au site z ∈ Zd par ω(z, ·) = ω(z, z + e)e∈Zd,|e|=1 . Cette formulation peutparaître compliquée mais se représente assez simplement, on s'imagine qu'en chaque sitez on choisit une mesure probabilité sur les voisins de z, chargeant z + e avec un poidsω(z, z + e). Un exemple d'environnement est fourni dans la gure 1.1.

L'élément ω ∈ Ω est appelé environnement et c'est dans celui-ci que la marchealéatoire va se déplacer. Etant donné un environnement ω on appelle marche aléatoiredans l'environnement ω partant de x, la chaîne de Markov (Xn)n≥0 dénie par X0 = xP xω -p.s. et

pour tout n ≥ 0 et x ∈ Zd, P xω [Xn+1 = x+ e | Xn = x] = ω(x, x+ e),

une loi que nous abrégerons dans le cas x = 0 par Pω := P 0ω .

2

2. FORMULATION MATHÉMATIQUE

0

x

ω(y, y + e2)

ω(y, y + e1)ω(y, y − e1)

ω(y, y − e2)

ω(x, x + e2)

ω(x, x + e1)

ω(x, x− e2)

ω(x, x− e1)

y

Fig. 1.1 Exemple d'environnement

Cette loi Pω est appelée quenched, un terme signiant trempée qui provient de lamétallurgie. Il est également utilisé en physique statistique pour désigner un système àdésordre xé, par exemple la position de particules magnétisées dans un alliage neutre.Cette loi a l'avantage d'être markovienne mais n'est pas invariante par translation cargénériquement ω ne l'est pas. On considère maintenant un environnement aléatoire cequi revient à mettre une probabilité P sur l'espace Ω. Ce qui nous permet d'introduirela loi P de la marche aléatoire moyennée sur l'environnement qui est dénie commeproduit semi-direct de P et Pω i.e.

P(·) =

∫Pω(·)dP(ω).

Cette loi est appelée annealed, ou également recuite dans le vocabulaire métal-lurgique. Elle n'est jamais markovienne si P n'est pas un dirac, i.e. si l'environnementest réellement aléatoire, mais si l'environnement est invariante par translation (ce quiest commun) alors elle est invariante par translation.

Donnons deux types d'environnement communément utilisés : les marches aléatoires en environnements aléatoires où les probabilités de transi-tions sont i.i.d., i.e. P = µ⊗Zd où µ est une probabilité sur e ∈ Zd, |e| = 1,

les marches aléatoires en conductances aléatoires dont les probabilités de transi-tions sont données pour tout x ∈ Zd et e ∈ Zd avec |e| = 1 par,

Pω(x, x+ e) =cω(x, x+ e)∑

e′∈Zd, |e′|=1 cω(x, x+ e′),

où les cω(x, x+ e′) sont des variables aléatoires i.i.d.. Ce modèle est naturel car il

3


permet de dénir des marches aléatoires réversibles en milieux aléatoires et à cetitre est fortement lié à la théorie des réseaux électriques, voir [30] et [70].

3 Organisation de la thèse

Notre objectif n'est en aucun cas de faire une introduction aux M.A.M.A. dans uncadre complétement général, nous renvoyons le lecteur aux ouvrages de références [100]et [104] pour plus de généralités. Cependant les modèles que nous avons étudiés sontassez divers dans leur formulation et il nous sera donc nécessaire les présenter rapide-ment.

Nous allons rapidement expliquer les liens qu'il existe entre les modèles étudiésdans cette thèse. Ce qui les unit est leur mode de fonctionnement, il s'agit de mod-èles anisotropiques, où la particule est poussée dans une direction particulière, et qui deplus présentent des zones qui ralentissent fortement la marche.

Le modèle constituant le point de départ de cette thèse est celui de la marche aléatoirebiaisée sur le cluster de percolation. Il s'agit d'un modèle très important, en eet ilest à la fois naturel physiquement, ce qui explique l'intérêt que les physiciens lui ontporté, et il présente des questions mathématiques intéressantes et diciles à traiter. Uneprésentation plus précise de ce modèle est faite à la Section 4.3.

Ce modèle étant apparemment hors de portée au début de la thèse, je me suis tournévers l'étude d'un modèle proche sous beaucoup d'aspects mais plus simple à analyser. Ils'agit de la marche aléatoire biaisée sur un Galton-Watson avec des feuilles, sur lequeldes comportements similaires à ceux qu'on observe sur le cluster de percolation ont étédémontré mathématiquement. En étudiant ce modèle, j'ai très vite trouvé de fortes sim-ilarités avec les modèles uni-dimensionnels. En eet le biais pousse la marche dans unedirection privilégiée et la trajectoire vue de loin est essentiellement uni-dimensionnelle.De plus les résultats plus ns obtenus par les physiciens étaient très similaires à ce quel'on avait obtenu sur Z, il était donc naturel d'aller travailler également sur les marchesaléatoires en milieu aléatoire sur Z.

Comme souvent en recherche, on part d'un modèle complexe et l'on cherche desmodèles plus simples ou mieux compris avec lesquels faire des parallèles. Le déroulementde cette thèse ne fait pas exception à la règle. Cependant pour la présentation desrésultats qui permettent de replacer notre thèse dans le contexte nous allons suivre,autant que possible, l'ordre historique d'apparition des résultats. Ainsi nous étudieronsdes modèles de plus en plus diciles. Plus précisément nous allons présenter les modèlesen fonction de la dimension

dans la section 1 on présentera les M.A.M.A. sur Z, dans la section 2 on présentera les M.A.M.A. sur les arbres, dans la section 3 on présentera les M.A.M.A. sur Zd avec d ≥ 2.Nos résultats sont présentés dans le chapitre 3 et leurs preuves sont incluses (en

4

3. ORGANISATION DE LA THÈSE

anglais) dans les chapitres 4, 5, 6.

5


6

2Quelques modèles d'importance

1 La marche aléatoire en environnement aléatoire uni-

dimensionnelle

1.1 Le modèle et son histoire

La marche aléatoire en environnement aléatoire uni-dimensionnelle est le modèle leplus simple de M.A.M.A.. Il a été introduit en 1967 par le biophysicien Chernov [19]pour comprendre les phénomènes de duplication des brins d'A.D.N.. Plus récemmentde nouveaux résultats sont apparus en biologie en lien avec des expériences de micro-manipulation des brins d'A.D.N., citons par exemple Lubensky et Nelson [64].

Ce modèle apparait également en métallurgie, comme l'indique les liens de vocabu-laire étroits. En eet, en 1972, Temkin le réutilise pour étudier la cinétique des transitionsde phase dans les alliages. Finalement pour des résultats obtenus en physique théoriquesur ce modèle, nous renvoyons le lecteur à Le Doussal, Monthus et Fisher [61].

En raison de la simplicité du graphe portant l'environnement aléatoire les notationssont particulièrement simples. On se donne ω = (ωx)z∈Z une famille de variables aléa-toires indépendantes et identiquement distribuées à valeurs dans ]0, 1[. Les probabilitésde transition de notre chaîne de Markov sont données pour n ≥ 0 et x ∈ Z par

Pω[X0 = 0] = 1 et Pω[Xn+1 = x+ 1 | Xn = x] = ωx = 1− Pω[Xn+1 = x− 1 | Xn = x].

7

CHAPITRE 2. QUELQUES MODÈLES D'IMPORTANCE

Typiquement on représente l'environnement comme dans la gure 2.1.

xx− 1 x + 1

1− ωx ωx

Fig. 2.1 Exemple d'environnement sur Z

On supposera dans la suite que l'environnement ω n'est pas déterministe. La chaîne(Xn)n≥0 n'est donc pas markovienne, de plus dans ce cas la loi annealed est invariante partranslation. Les résultats et les méthodes classiques concernant les marches aléatoires nepeuvent se transposer. La palette de phénomènes apparaissant en dimension 1 est déjàextrêmement riche. Elle a certainement contribué au fort intérêt que la communautéprobabiliste a porté aux M.A.M.A..

Avant de rentrer dans la description des résultats liées à ce modèle, précisons qu'ilexiste d'autres modèles uni-dimensionnels qui ont été étudiés. Par exemple, on peutconsidérer des probabilités de transitions ergodiques, réversibles ou bien des marchesqui ne sont pas à plus proches voisins. On renvoit par exemple le lecteur à [60], [3], [74].

1.2 Transience-récurrence et loi des grandes nombres

Les premiers résultats mathématiques sont apparus en 1975, Solomon [95] obtientun critère de récurrence-transience pour la M.A.M.A. uni-dimensionnelle. Il va mêmejusqu'à une loi des grands nombres. Contrairement au cas de l'environnement détermin-iste, ce n'est pas ici la dérive de la marche, i.e. E[X1], qui apparait dans la loi des grandsnombres. La variable aléatoire qui s'avère centrale est

ρ0 =1− ω0

ω0

.

On supposera que la quantité E[ln ρ0] est bien dénie (éventuellement innie).

Théorème 1.1 (Solomon-1975). On a deux cas.

1. Si E[ln ρ0] < 0 (resp. >0) alors la marche est transiente et on a

limn→∞

Xn =∞ (resp. −∞) P-p.s..

2. SI E[ln ρ0] = 0 alors la marche est récurrente et

lim supn→∞

Xn =∞ et lim infn→∞

Xn = −∞ P-p.s..

La loi des grands nombres se formule de la manière suivante

8

1. LA MARCHE ALÉATOIRE EN ENVIRONNEMENT ALÉATOIREUNI-DIMENSIONNELLE

Théorème 1.2 (Solomon-1975). On a

limn→∞

Xn

n= v, P-p.s.,

avec

v =

1−E[ρ0]1+E[ρ0]

si E[ρ0] < 1E[1/ρ0]−1E[1/ρ0]+1

si E[1/ρ0] < 1

0 si 1/E[1/ρ0] ≤ 1 ≤ E[ρ0].

Ce théorème constitue le tout premier résultat signicatif et on peut déjà y voir lesphénomènes de ralentissement qui constitue le coeur de cette thèse. Les deux pointscentraux sont les suivants

1. on a |v| < |E[X1]|,2. il existe des régimes où la M.A.M.A. est transiente dans une direction avec cepen-

dant une vitesse nulle.

Le premier phénomène montre qu'en dimension 1, la marche est eectivement ralentiepar l'aléatoire dans l'environnement. Comprendre la vitesse d'une M.A.M.A. est enfait une question très épineuse. Nous verrons plus tard, voir Section 3, qu'il existe descomportements plus riches en dimension d ≥ 2 et que cette propriété n'est pas conservée.En dimension supérieure le seul outil pour étudier la vitesse d'une M.A.M.A. est de larelier à celle d'une marche aléatoire dans un environnement déterministe compliqué.

Le deuxième phénomène est annonciateur de l'existence de nouveaux régimes quin'existaient pas dans le cadre des marches aléatoires classiques. Cela va de paire avecl'apparition de plusieurs nouvelles questions. En particulier il parait naturel de se de-mander l'ordre de grandeur des mouvements de le marche. Peut-on trouver une fonctionsimple f(n) (par exemple polynomiale) telle que (Xn/f(n))n≥0 converge en loi ? formeune famille tendue ? etc.

Un régime similaire, de transience directionnelle à vitesse nulle, existe pour lesmarches aléatoires en conductances aléatoires sur les arbres et en dimension supérieureégalement. Il constitue en quelque sorte le véritable noyau de cette thèse.

1.3 Le cas récurrent : le potentiel de Sinaï

Avant de revenir plus en détail sur le régime transient à vitesse nulle, nous allonsintroduire l'objet qui semble être le plus pertinent pour l'analyse de la M.A.M.A. uni-dimensionnelle. Il s'agit du potentiel dit de Sinaï qui fut introduit en 1982, voir [94],et qui est déni de la manière suivante

V (x) :=

∑x

i=1 ln ρi, si x ≥ 1,

0, si x = 0,

−∑0i=x+1 ln ρi, si x ≤ −1,

9


D'un point de vue physique cette quantité correspond à une énergie potentielle, d'oùle nom, et donne une interprétation visuelle des endroits où la marche aura tendance àrester bloquée. Mathématiquement on pourra noter les deux faits suivants

V (x) est une marche aléatoire (à valeurs réelles) de pas i.i.d. de loi ln ρ0, V (x) permet de dénir facilement une mesure invariante pour la marche dansl'environnement ω en posant π(x) = e−V (x) + e−V (x−1).

Sous des hypothèses de moments sur les sauts du potentiel, ce dernier se comporteessentiellement comme un mouvement brownien. La gure 2.2 correspond à un exempletypique de potentiel.

V (x)

x

0Xn

Fig. 2.2 Exemple de potentiel dans le cas transient vers ∞On peut ainsi facilement expliquer le résultat de transience obtenu par Solomon, si

E[ln ρ0] < 0 alors V (x) va aller vers −∞ seulement en +∞. Or comme la marche estattirée par les zones qui sont chargées par la mesure invariante, i.e. de bas potentiel,elle va naturellement partir vers +∞. En quelque sorte le potentiel de Sinaï permetd'expliquer l'importance de la variable aléatoire ln ρ0.

Bien évidemment l'utilisation de ce potentiel va bien au delà de cette interprétationintuitive du critère de transience-récurrence. Originellement, il a permis de démontrerdans [94] le résultat suivant.

Théorème 1.3 (Sinai-1982). Supposons que P[ω0 ∈ [ε, 1 − ε]] = 1 pour ε > 0 et queE[ln ρ0] = 0, alors

Xn

ln2 n

loi−→ b∞,

10

1. LA MARCHE ALÉATOIRE EN ENVIRONNEMENT ALÉATOIREUNI-DIMENSIONNELLE

où b∞ est non-dégénérée et non gaussienne qui ne dépend que de l'environnement.

L'expression explicite de la loi de b∞ a ensuite été obtenue indépendamment parKesten [57] et Golosov [44].

On notera que l'hypothèse P[ω0 ∈ [ε, 1 − ε]] = 1 pour ε > 0, dite d'uniforme el-lipticité, n'est pas seulement une hypothèse simplicatrice pour éviter des problèmesd'intégrabilité. Il permet en eet d'approximer le potentiel vu de loin par un mouve-ment brownien. Pour des potentiels plus irréguliers, le potentiel se comporte comme unprocessus stable d'indice α. Kawazu, Tamura et Tanaka [56] montrent que le déplace-ment de la marche est alors en lnα(n).

1.4 Le cas transient à vitesse nulle

Revenons-en maintenant au cas de la transience directionnelle à vitesse nulle. Cerégime fut étudié rapidement, le premier résultat remontant à Kesten, Kozlov et Spitzer [59].Ce résultat est important dans l'histoire des M.A.M.A., à tel point qu'on parle souventde régime Kesten-Kozlov-Spitzer ou bien régime K.K.S.. Il a connu ces dernières an-nées un regain d'intérêt sous l'impulsion du travail de Enriquez, Sabot et Zindy, voir [35]et [36].

Théorème 1.4 (Kesten, Kozlov, Spitzer - 1975). Supposons que la famille de v.a.i.i.d. ω :=(ωi, i ∈ Z) vérie

1. −∞ ≤ E[ln ρ0] < 0,

2. il existe 0 < κ < 1 tel que E [ρκ0 ] = 1 et E[ρκ0 ln+ ρ0

]<∞,

3. la distribution de ln ρ0 n'est pas concentrée sur un réseau,

alorsτ(n)

n1/κ

loi−→ Sκ, et Xn

nκloi−→ Lκ,

oùloi−→ désigne la convergence en loi sous la mesure P, Scaκ une loi stable complètement

assymétrique d'indexe κ et Lκ une loi de Mittag-Leer d'indice κ.

Dans le même article il a également été démontré un comportement en n/ lnn dans lecas où E[ρ0] = 1 ainsi que des résultats de uctuations dans le cas balistique à variationsnon-gaussiennes.

Ce résultat a été rané dans [35] et [36] au sens où la description des lois limitesest plus précises. De plus les méthodes de démonstration mises en oeuvre et utilisant lepotentiel semblent assez robustes. Elles ont fourni une grande source d'inspiration pourdeux des nouveaux résultats contenus dans cette thèse, voir ([39]) et ([8]).

Concernant les théorèmes limites quenched la situation est compliquée, on peut enfait démontrer qu'il n'y a pas convergence en loi [79].

11


1.5 Principes de grandes déviations

Un dernier type de résultat qu'il convient de citer, car il complète en quelque sorteles résultats obtenus précédemment, concerne les grandes déviations. Par exemple dansle cas d'une marche transiente à vitesse positive, on cherche à savoir à quelle vitesseva décroitre la probabilité que P[Xn/n ≤ c] pour 0 < c < v où v désigne la vitesseasymptotique.

Nous citerons le premier résultat concernant un principe de grandes déviations(P.G.D.) sous la loi quenched obtenu en 1994 par Greven et den Hollander [45]. Cerésultat fut démontré via des méthodes d'homogénéisation et a ceci de surprenant quela fonction de taux obtenue dans le P.G.D. est déterministe. Ce résultat fût complété parComets, Gantert et Zeitouni dans [20] qui grâce à des méthodes diérentes ont obtenuun P.G.D. quenched et annealed. De nombreuses propriétés sur les fonctions de tauxsont obtenues, ce qui fait de cet article un recueil complet de résultats sur les grandesdéviations pour la M.A.M.A. uni-dimensionnelle. Le cas nestling (voir [100] pour ladénition précise) reste cependant partiellement laissé ouvert. En eet la fonction detaux obtenue est nulle aussi bien dans le cas quenched que annealed sur l'interval [0, v],ce qui nous indique seulement que la décroissance est sous-exponentielle.

D'autres résultats de grandes déviations peuvent-être trouvés dans [26], [27], [40],[41], [81] and [82].

2 Marches aléatoires en milieu aléatoire sur des arbres

2.1 Modèle

La M.A.M.A. en dimension 1 étant essentiellement bien comprise, la recherche s'estportée vers d'autres modèles plus complexes à analyser. Trois propriétés étaient à l'o-rigine de la simplicité d'analyse du modèle

1. la marche sur Z est systématiquement réversible, car Z ne contient pas de cycle,

2. pour aller d'un point x à un point y, la marche doit passer par tous les points de]x, y[,

3. il n'y a, dans le cas transient, qu'une seule manière de partir à l'inni.

En passant directement à l'étude des M.A.M.A. dans Zd avec d ≥ 2, nous perdonstoutes ses propriétés et cela explique les dicultés éprouvées pour prouver de nouveauxrésultats. Les mathématiciens se sont donc penchés sur l'étude d'un modèle intermédi-aire, celui des marches aléatoires sur les arbres, Dans ce cas, seule la troisième propriétécitée au-dessus est perdue. On peut également noter que le potentiel, provenant du car-actère réversible de la marche, existe mais ne peut être représenté en terme de marchealéatoire comme en dimension 1.

12

2. MARCHES ALÉATOIRES EN MILIEU ALÉATOIRE SUR DESARBRES

Jusqu'ici nous avons seulement fait référence à des marches aléatoires qui vivent surun graphe sous-jacent xé, par exemple Zd. Dans le cas des arbres, il est commun deconsidérer des marches sur un arbre de Galton-Watson qui est lui-même aléatoire, surlequel on peut également rajouter un environnement aléatoire.

Précisons le modèle le plus communément étudié. Pour cela nous avons besoin

1. d'une suite de v.a.i.i.d. (Z(i))i≥0 suivant une loi de reproduction P [Z(1) = k] = pksurcritique i.e. telle que m =

∑k kpk ∈ (1,∞),

2. et d'une suite de v.a.i.i.d. (Ai)i≥0 à valeurs dans R∗+,

où nous excluons évidemment le cas p1 = 1 qui correspond à considérer une M.A.M.A. uni-dimensionnelle.

On xe au départ deux sommets −→r et r reliés par une arête et on attribue unegénération à −→r (resp. r) qui est −1 (resp. 0). On construit alors récursivement l'arbrealéatoire de la manière suivante : lorsque la génération n est construite nous pouvonsnoter les sommets de cette génération x1, . . . , xk. Pour un de ses sommets xi nous ajou-tons Z(i) (où la variable utilisée est indépendante de celles utilisées précédemment dansla construction) enfants chacun étant relié à xi par une arête, de plus chacun de sesenfants est aecté d'une marque aléatoire tirée indépendament suivant la loi Ai.

Ce processus fournit un Galton-Watson où les sommets sont marqués, nous le noteronsT. Notre mesure sur l'environnement est alors la loi de T conditionnée à être inni, cequi est un événement de probabilité positive car nous avons supposé que notre Galton-Watson était surcritique.

Conditionnellement à la donnée d'un tel arbre T(ω) la marche aléatoire est déniede la manière suivante : pour tout y ∈ T et tout n ≥ 0, on pose

Pω[X0 = r] = 1 et Pω[Xn+1 = z | Xn = y] = ω(y, z),

où ω(−→r , r) = 1 et pour tout x ∈ T \ −→r qui admet x1, . . . , xk comme enfants

1. ω(x, xi) = A(xi)

1+Pki=1 A(xi)

, pour i = 1, . . . , k,

2. ω(x,−→x ) = 1

1+Pki=1 A(xi)

, où −→x est le père de x,

3. ω(x, y) = 0, sinon.

Ce nouveau type de M.A.M.A. se divise essentiellement en deux catégories biendistinctes, le cas où p0 > 0 et le cas où p0 = 0. Le premier se révèle plus dicile àanalyser pour diverses raisons. Nous allons simplement citer le fait que dans ce casnous perdons toute invariance par translation annealed car c'est le seul cas où notremesure sur l'environnement correspond à un Galton-Watson conditionné à survivre. Enparticulier ce n'est pas parce qu'un arbre est transient que la descendance de n'importequel point est un arbre transient (ce qui est le cas si p0 = 0). Cela complique l'analysedu modèle.

13


2.2 Transience-récurrence

Les questions de transience-récurrence sur les arbres ont été énormément étudiéesau début des années 1990. Les outils développés à cette époque (voir [77] ou [70]) enparticulier les liens avec les réseaux électriques [30], se révèlent relativement robustes.

Dans le cas du modèle présenté ci-dessus nous avons, voir [66],

Théorème 2.1 (Lyons, Pemantle - 1992). Nous avons le critère suivant

1. si inft∈[0,1] E[At] > 1/m alors Xn est transiente,

2. si inft∈[0,1] E[At] < 1/m alors Xn est récurrente.

Lorsque le Galton-Watson est déterministe, Menshikov et Petretis [75] obtiennent uncritère en utilisant un lien entre les M.A.M.A. et la cascade multiplicative de Mandelbrot.

Cette thèse est essentiellement concernée par le régime transient, pour cela nousn'allons pas entrer plus dans les détails des résultats sur le régime récurrent. Nousrenvoyons seulement le lecteur aux articles de Hu et Shi [52], [53] qui montrent que lecas récurrent est également intéressant car il présente plusieurs régimes diérents deceux obtenus en dimension 1.

2.3 Loi des grands nombres

Dans le cas transient la première question qui apparait est de savoir si on part àl'inni à vitesse strictement positive ou non. En d'autres termes on cherche à démontrerune loi des grands nombres.

On a tout d'abord obtenu qu'il existe v ≥ 0 déterministe tel que

limn→∞

|Xn|n

= v, P-p.s..

Ce résultat à été démontré par Gross [49] dans le cas où le Galton-Watson estdéterministe et par Lyons, Pemantle et Peres [68] dans le cas où A est déterministe,leurs arguments s'étendant facilement au cas général.

La question est maintenant de savoir si v > 0. Elle fut d'abord traitée dans le cas dela marche aléatoire biaisée sur un Galton-Watson, i.e. dans le cas où A est déterministe.Le premier résultat fut obtenu dans [68]

Théorème 2.2. Soit Xn une marche aléatoire biaisée avec un biais λ, i.e. A = λ p.s.telle que Xn soit transiente, i.e. λ > 1/m, alors

1. si p0 = 0 alors Xn admet une vitesse positive i.e. v > 0,

2. si p0 > 0 alors Xn admet une vitesse positive si, et seulement si, λ < 1/f ′(q) oùf(x) =

∑k≥0 pkz

k et q est l'unique point xe de f(·) diérent de 1.

14

2. MARCHES ALÉATOIRES EN MILIEU ALÉATOIRE SUR DESARBRES

Le deuxième phénomène peut paraître surprenant à première vue. Il est en fait dû àun ralentissement de la marche qui reste coincée dans les pièges formés par les feuilles,i.e. les zones de l'arbre qui possèdent une descendance nie. Nous reviendrons sur cephénomène dans la Section 2.

Il convient de noter que des questions plus précises concernant la vitesse sont ex-trêmement complexes et pour l'instant majoritairement ouvertes. Nous renvoyons à [69]pour une présentation assez détaillée de la question suivante : est-ce que la vitesse estune fonction croissante du biais λ si p0 = 0 ? Malgré l'étonnante simplicité de l'énoncédu problème et le caractère naturel du résultat, elle demeure ouverte depuis plus dedix ans. Cette parenthèse montre la diculté à obtenir des expressions explicites, oùplus généralement des propriétés nes, sur la vitesse asymptotique. A ce jour seul le casλ = 1, i.e. sans biais, est compris [67].

Revenons-on maintenant au cas plus général où A est aléatoire. Nous allons présenterun résultat obtenu dans le cadre p0 = 0 par Aidekon, qui trouve un critère de ballisticitéet identie l'exposant de renormalisation de la marche dans le cas sous-balistique. Posons

Λ := Lebt ∈ R, E[At] ≤ 1/q1

,

qui est arbitrairement xé à Λ :=∞ dans le cas q1 = 0. On a alors, voir [2], le théorèmesuivant.

Théorème 2.3. Si inft∈[0,1] E[At] > 1/m, i.e. dans le cas transient, alors

si Λ > 1, alors Xn admet une vitesse positive, si Λ < 1, alors Xn admet une vitesse nulle et

limn→∞

ln(|Xn|)lnn

= Λ, P-p.s..

On obtient donc également un régime de transience directionnelle avec vitesse nulleavec des variations d'ordre polynomiale comme dans le cas uni-dimensionnel.

2.4 Principes de grandes déviations

L'étude des M.A.M.A. sur les arbres étant assez récente, beaucoup de résultatsrestent encore à démontrer et cette partie resterait à remplir. Nous tenons juste à citerdeux des principaux résultats existants. Le premier résultat remonte à Dembo et al. [25],où les auteurs démontrent un P.G.D. dans les cas quenched et annealed dans le cas desmarches aléatoires biaisées (i.e. A déterministe). Ils obtiennent en particulier égalité desdeux fonctions de taux. Plus récemment le cas p0 = 0 et A aléatoire a été traité dans [1].

15


3 Marches aléatoires en environnements aléatoires sur

Zd avec d ≥ 2

Nous arrivons maintenant au cadre qui aura suscité le plus d'intérêt ses dix dernièresannées mais aussi sur lequel le moins de choses sont bien comprises. Nous allons séparerles M.A.M.A. sur Zd en deux parties. En eet il existe deux grandes catégories deM.A.M.A. qui ont été étudiées jusqu'à ce jour.

1. Les marches aléatoires en environnements aléatoires elliptiques, i.e. avec P[ω(x, x+e) > 0] = 1 pour tout x ∈ Zd et e ∈ Zd avec |e| = 1.

2. Les marches aléatoires sur un cluster de percolation qui sont dénies de manièreà être réversible et par nature ne sont pas elliptiques.

Cette distincition est à mettre en lien avec la disjonction de cas p0 = 0, p0 > 0faite dans la section précédente dans le cadre des marches aléatoires sur les arbres. Nousallons commencer par traiter les marches aléatoires en environnements aléatoires.

3.1 Le modèle

Nous allons présenter le modèle quitte à faire une redite de la partie introductivepour rappeler les notations du modèle.

On se donne µ une loi elliptique sur le simplexe de R2d+ , i.e. une loi de probabilité sur

S∗ =

(p1, . . . , p2d) ∈ (0, 1)d, tel que2d∑i=1

pi = 1.

On choisit ensuite un environnement aléatoire ω ∈ (S∗)Zd suivant la loi P = µ⊗Zd .La marche dans ω = ((ω(x, e))e∈Zd,|e|=1)x∈Zd a alors pour loi Pω dénie par

Pω[X0 = 0] = 1,

et

Pω[Xn+1 = Xn + e | X0, . . . , Xn] = ω(Xn, e), e ∈ Zd, |e| = 1.

Nous insistons sur le fait que nous avons supposé que l'environnement est elliptique,i.e. que les probabilités de transition sont toujours positives. Une hypothèse supplé-mentaire sera faite de temps en temps, elle est dite d'uniforme ellipticité, et revient àsupposer que

il existe ε > 0 tel que, P[ω(0, e) > ε] = 1, pour tout e ∈ Zd avec |e| = 1. (3.1)

16

3. MARCHES ALÉATOIRES EN ENVIRONNEMENTS ALÉATOIRESSUR ZD AVEC D ≥ 2

3.2 Transience-récurrence

Le premier résultat lié aux questions de transience-récurrence pour les M.A.M.A. endimensions supérieures remonte à Kalikow [55]. Il démontre une loi du 0−1 et introduitla notion d'environnement de Kalikow dont nous reparlerons dans la Section 3.2.

Il introduit la notion de transience directionnelle, que nous avons évoqué précédem-ment sans en donner une dénition précise. Nous dirons qu'une marche est transientedans la direction ` ∈ Sd−1 (où Sd−1 désigne la sphère euclidienne de Zd) sur l'événement

A` =

limn→∞

Xn · ` = +∞.Le résultat originel de Kalikow utilisait l'uniforme ellipticité, une hypothèse aaiblie

plus tard par Merkl et Zerner dans [106]

Théorème 3.1 (Kalikow-1981). Toute marche aléatoire en environnement aléatoireelliptique vérie

P[A` ∪ A−`] = 0 ou 1.

La question L'événement A` satisfait-il une loi du 0 − 1 a été posée dans [55],néanmoins la réponse à cette question n'a pas été trouvée, excepté en dimension d = 2,voir [106] ou [105] pour une preuve simpliée de Zerner. Nous renvoyons aussi le lecteurau travail de Simenhaus [93] pour plus de résultats sur cette question.

Concernant la question de transience-récurrence proprement dite, nous devons pourl'instant nous contenter de savoir que le problème est bien posé au sens où

Théorème 3.2 (Kalikow - 1981). L'événement (Xn)n≥0 est récurrent sous Pω est deprobabilité 0 ou 1.

Ce résultat est dû au fait que cet événement fait partie de la tribu de queue de l'en-vironnement et le théorème est donc une conséquence de la loi du 0− 1 de Kolmogorov.

Nous allons fournir un critère de transience directionnelle qui remonte à Kalikow [55].La démonstration de ce critère fait appel à un outil intéréssant qui permet de comparerdes propriétés annealed d'une M.A.M.A. à des propriétés d'une chaîne de Markov. Ils'agit des environnements de Kalikow, il s'agit un des éléments clefs pour la preuve denotre résultat principal dans [38], voir Section 1. Nous allons donc le présenter en détail.

Environnement de Kalikow

Pour U ⊂ Zd, on note TU = infn ≥ 0, Xn /∈ U. Le but de l'environnement estde relier la loi annealed, qui n'est pas markovienne, à un environnement markovien. Sion suppose de plus que U est connexe et contient 0, on dénit la chaîne de Markov quiadmet U ∪ ∂U comme espace d'état et dont les probabilités de transition sont donnéspar

ωU(x, x+ e) =E[gωU(0, x)ω(x, e)]

E[gωU(0, x)], x ∈ U, |e| = 1,

17


ωU(x, x) =1, x ∈ ∂U,où nous avons utilisé la notation gωU(·, ·) pour désigner

gωU(x, y) = Eω

[ TU∑i=0

1Xn = y].

Un lien entre les M.A.M.A. et la marche dans l'environnement de Kalikow, i.e. donnéepar les probabilités de transition pU(·, ·), est énoncé dans le théorème suivant

Théorème 3.3 (Kalikow - 1981). Notons P0,U la loi d'une marche aléatoire issue de 0

donnée par les probabilités de transition ωU . Si P0,U [TU <∞] = 1 alors P0[TU <∞] = 1,

de plus XTU a même loi sous PU et P0.

Remarque 3.1. D'autres propriétés existent, en particulier on notera que E[gωU(0, x)] =gωUU (0, x).

Cet environnement peut-être utilisé en introduisant la dérive de Kalikow

dU(x) = Ex,U [X1 −X0], x ∈ U ∪ ∂U,qui permet de dénir le critère de Kalikow relatif à ` ∈ Sd−1

infU,x∈U

dU(x) · ` = ε(`, µ) > 0.

Cette condition peut-être dicile à vérier en général en dimensions supérieures.Des critères alternatifs plus simple à vérier existent, voir [102]. Elle permet d'énoncerle premier résultat de transience directionnelle

Théorème 3.4 (Kalikow -1981). Supposons avoir une M.A.M.A. dans Zd avec qui estuniformément elliptique et qui vérie le critère de Kalikow relatif à ` ∈ Sd−1, alors

limXn · `→∞, P-p.s..

Nous allons tout de suite voir que cette condition est en réalité bien plus forte.

3.3 Existence et étude de la vitesse

Après ces premiers résultats de transience directionelle de Kalikow, peu de résultatssont apparus pendant une période assez longue. Dans cette partie, nous ne cherchons enaucun cas à être exhaustif concernant la litérature concernant la loi des grands nombres.Nous allons simplement évoquer la premier résultat du genre qui est dû à Sznitman etZerner en 1999 [102]. Ce résultat est important au sens où il a relancé les recherches surles M.A.M.A. en dimensions supérieures.

18

3. MARCHES ALÉATOIRES EN ENVIRONNEMENTS ALÉATOIRESSUR ZD AVEC D ≥ 2

Ce théorème utilise de manière centrale deux outils. Le premier est celui de l'ex-istence d'une structure de régénération, nous renvoyons à l'article original [102] pourune description précise. L'idée est de découper l'environnement et la marche en blocsi.i.d., il sut ensuite de mesure l'avancée moyenne et le temps moyen passé dans untel bloc pour obtenir une loi des grands nombres pour la marche. Le deuxième outilqu'ils utilisent est le lien existant entre les M.A.M.A. et les marches aléatoires dans lesenvironnements de Kalikow.

Le résultat principal de [102] de la manière suivante

Théorème 3.5 (Sznitman, Zerner - 1999). Supposons avoir une M.A.M.A. dans Zd avecqui est uniformément elliptique et qui vérie le critère de Kalikow relatif à ` ∈ Sd−1,alors il existe v déterministe tel que

Xn

n→ v, P-p.s.,

et de plus v · ` > 0.

La condition de Kalikow caractérise exactement les marches balistiques en dimen-sion 1. En dimensions supérieures ce critère s'avère plus dicile à vérier. On noteral'utilisation de ce critère par Enriquez et Sabot dans [33] pour montrer que certainesmarches aléatoires dans les environnement de Dirichlet sont balistiques.

D'autres recherches pour caractériser la classe des marches balistiques ont permisd'obtenir plusieurs autres critères pour obtenir une vitesse positive. Le lecteur intéressépourra consulter les travaux de Sznitman, voir [100], sur les conditions (T ) et (T ′) pourplus d'informations.

Il n'existe quasiment aucun résultat plus précis sur le comportement de la vitesse, carla tâche est encore plus dicile que sur les arbres où les résultats sont déjà très rares.Nous citerons quand même un résultat de Sabot [90] qui étudie des environnementsfaiblement perturbés, i.e. du type ω(z, e) = p0(e)+εξ(z, e) où les ξ(z, e) sont i.i.d. et ε estsusamment petit. Il obtient un développement asymptotique de la vitesse en fonctionde ε. En particulier il démontre qu'en dimension supérieure il est possible que la marchesoit accélérée par l'environnement aléatoire au sens où la vitesse asymptotique est plusgrande que la dérive moyenne. Sa méthode d'étude repose sur l'étude de l'environnementde Kalikow associé, ne reviendrons plus en détails sur cela dans la Section 3.2.

3.4 Autres résultats

Théorème central limite

Il existe des critères pour obtenir des théorèmes du type théorème central limiteannealed. Nous renvoyons le lecteur à [98] (Théorème 3.3) et au livre [100] (Chapitre 4)pour plus de précisions.

Il existe également des principes d'invariance quenched, voir [85], [86], [87] et [16].

19


Grandes déviations

Une nouvelle fois les questions des grandes déviations sont étudiées. On pourra trou-ver des plus amples précisions dans le livre de Sznitman [98] (Théorème 3.4) et dans unarticle récent de Berger [12].

4 Marches aléatoires sur des clusters de percolation

Encore une fois, il existe plusieurs types de marches aléatoires que l'on peut dénirsur des clusters de percolation. Nous allons ici présenter la marche aléatoire simple etla marche aléatoire biaisée sur le cluster de percolation.

Ces modèles se diérentient fortement des marches aléatoires en environnementsaléatoires principalement pour deux raisons

1. ils ne sont pas elliptiques,

2. ils sont réversibles.

Ce dernier point rend plus facile l'analyse de la marche sous l'environnement quenched,la perte de l'invariance par translation sous la loi annealed peut être compensée par laréversibilité du modèle. Mais il est clair que les méthodes de démonstration vont forte-ment diérer.

4.1 La percolation par arêtes

Il ne s'agit pas ici de présenter toute la théorie de la percolation, nous allons nous con-tenter d'introduire des notations et le minimum nécessaire pour comprendre la présen-tation des théorèmes. Pour de plus amples informations sur cet immense domaine, onpourra trouver une introduction dans l'ouvrage de Grimmett [46].

Nous allons seulement présenter la percolation par arêtes sur Zd. On xe un paramètrep ∈ [0, 1]. Nous voulons étudier le graphe aléatoire obtenu en enlevant chaque arête avecprobabilité p indépendamment de toutes les autres arêtes. Mathématiquement, on noteΩ = 0, 1Ed où Ed désigne les arêtes de Zd, on dira qu'une arête est présente ou ou-verte (resp. absente ou fermée) si ω(e) = 1 (resp. ω(e) = 0). On munit cet espacede la mesure

Pp = (Ber(p))⊗Ed .

Dans ce graphe aléatoire nous avons naturellement une notion de connexité et lescomposantes connexes de ce graphe sont appelées des clusters. L'un des résultats fon-damentaux de percolation dont nous avons besoin est résumé dans le théorème suivant(voir Théorème 1.10 p.14 et Théorème 8.1 p.198 dans [46])

Théorème 4.1. Pour tout d ≥ 2, il existe pc(d) ∈ (0, 1) tel que,

1. pour p < pc, Pp-p.s. tous les clusters sont nis,

20

4. MARCHES ALÉATOIRES SUR DES CLUSTERS DE PERCOLATION

2. pour p > pc, Pp-p.s. il existe un unique cluster inni.

La première phase est appelée sous-critique et la deuxième sur-critique. Donc seule ladeuxième phase permet d'obtenir un graphe inni où des questions de types récurrence-transience pourront être étudiées. Nous allons donc nous restreindre à l'étude de cerégime.

On remarque que ce théorème implique que pour p > pc(d)

Pp[|K(0)| =∞] > 0,

où K(0) désigne le cluster de 0 et |K(0)| sa taille. Ainsi il est possible conditionner lecluster inni à passer par 0 en dénisant la mesure

Pp[ · ] = Pp[ · | |K(0)| =∞].

D'autres modèles existent, nous citerons l'étude des marches aléatoires sur le clustercritique [58] où les marches ralenties sur les clusters [23] et [83].

Dans la suite on xe p > pc(d) et par soucis de légéreté nous ometrons l'indice pdans Pp lorsqu'aucune confusion n'est possible. De plus ω désignera un environnementtiré sous la mesure P.

4.2 La marche aléatoire simple

Le modèle

La marche aléatoire simple sur un cluster de percolation est la chaîne de Markovdénie sur K(0) par

Pω[Xn+1 = x+ e | Xn = x] =ω([x, x+ e])∑

e′,|e′|=1 ω([x, x+ e′]),

qui est une quantité bien dénie car le dénominateur ne peut s'annuler sur K(0).

Résultats

Transience-Récurrence Par un argument classique de réseaux électriques (le théorèmede monotonicité de Rayleigh voir [30] ou [70]) la marche aléatoire simple sur un clusterde percolation est récurrente en dimension d ≤ 2, car intuitivement il y a moins defaçons de partir à l'inni. Ce qui n'est pas aussi clair est le fait que la marche restetransiente en dimension d ≥ 3, ce qui a été démontré dans [48].

Théorème 4.2 (Grimmett, Kesten, Zhang - 1992). La marche aléatoire simple sur lecluster de percolation de paramètre p > pc(d) est transiente si, et seulement si, d ≥ 3.

Nous noterons qu'il existe une preuve alternative de ce résultat par Benjamini, Pe-mantle et Peres [11].

Le fait que la percolation ne change pas la nature transiente (ou bien évidemmentrécurrente) est un fait relativement général [5].

21


Principe d'invariance Le principe d'invariance annealed remonte à [24] et à [47]pour obtenir que la variance limite est non nulle. Récemment de nombreux résultats ontpermis d'obtenir des principes d'invariance quenched, en commençant par Sidoravicius etSznitman [92] (pour d ≥ 4) puis plus récemment par Berger et Biskup [13] parallèlementà un travail de Mathieu et Piatnitski [72].

Des résultats similaires existent dans des modèles plus généraux de conductancesaléatoires i.i.d. [71]. Ces questions sont fortement liés à des questions d'isopérimétries.

Autres résultats Beaucoup d'autres questions sont liées à cette notion d'isopérimétrie.On renvoit le lecteur aux travaux de Rau [88] qui étudie le nombre de points visités parla marche aléatoire simple sur un cluster de percolation. D'autres questions concernentdes estimées sur le noyau de la chaleur [73] et [14].

4.3 La marche aléatoire biaisée

Le modèle

Il existe plusieurs manières de dénir une marche aléatoire biaisée sur un cluster depercolation, deux modèles ont été proposés l'un par Berger, Gantert et Peres [15] l'autrepar Sznitman [99]. Nous présenterons le deuxième qui est légèrement plus général car ilautorise un biais dans toutes les directions possible. On xe ~ ∈ Sd−1 et λ > 0, ce quinous donne un biais ` = λ~ de force λ et de direction ~. On peut alors dénir pour toutω, la marche aléatoire biaisée sur un cluster de percolation comme la chaîne de Markovdénie sur K(0) par

Pω[Xn+1 = x+ e | Xn = x] = pω(x, x+ e) =e`·eω([x, x+ e])∑

e′,|e′|=1 e`·e′ω([x, x+ e′])

.

Les résultats

Il existe plusieurs articles dans la littérature physique concernant ce modèle, voir [28]et [29]. Cependant mathématiquement ce modèle reste dicile à traiter, les seuls travauxexistant jusqu'à très récemment sont [15] et [99].

Ces deux articles démontrent essentiellement le même résultat qui est résumé dansle théorème suivant

Théorème 4.3 (Berger, Gantert, Peres - 2003 ; Sznitman -2003). On a

limn→∞

Xn · ` =∞, P-p.s..

De plus, il existe 0 < λl ≤ λu tel que

1. si λ < λl alors limn→∞Xn/n = v avec v · ` > 0,

22

4. MARCHES ALÉATOIRES SUR DES CLUSTERS DE PERCOLATION

2. si λ > λu alors limn→∞Xn/n = 0.

Ce théorème conrme partiellement les conjectures des physiciens qui de plus s'at-tendent à pouvoir énoncer le théorème avec λl = λu. Cependant mathématiquementnous sommes encore incapable d'exclure l'existence d'un possible régime intermédiaire.

23


24

3Présentation des résultats

Comme il a été évoqué dans l'introduction de la thèse, le modèle qui a motivé laplupart de mes recherches est celui de la marche aléatoire biaisée sur un cluster depercolation. Les deux questions majeures qui restaient en suspens, ayant à voir avec desphénomènes de ralentissement, sont

1. l'étude de la vitesse, i.e. déterminer le régime balistique et comprendre la dépen-dance de la vitesse vis-à-vis des paramètres,

2. l'identication de l'ordre de grandeur des uctuations de la marche.

Le premier type de problèmes est partiellement étudié dans la Section 1. Concer-nant la deuxième question, nous n'avons pas encore obtenu de résultats sur le clusterde percolation. Nous nous sommes donc tourné vers la marche aléatoire biaisée sur unGalton-Watson avec des feuilles, on pourra retrouver une présentation du résultat corre-spondant dans la Section 2. En étudiant ce problème nous avons eu l'occasion d'étudieren détails le comportement de la M.A.M.A. uni-dimensionnelle. Cela nous a permis d'é-tudier des questions de déviations modérées qui étaient restées ouvertes jusqu'ici, lerésultat correspondant étant présenté dans la Section 3.

25

CHAPITRE 3. PRÉSENTATION DES RÉSULTATS

1 Comportement de la vitesse sur le cluster de perco-

lation vis-à-vis des paramètres

Concernant l'étude de la dépendance de la vitesse vis-à-vis des paramètres du prob-lème, deux questions sont envisageables : la dépendance par rapport au biais ou parrapport au paramètre de percolation. En particulier on s'intéresse à d'éventuelles pro-priétés de monotonie.

La question la plus abordable semble être la première, car il existe déjà des résultatsqui, sur les arbres, vont dans ce sens (voir [18]) alors que concernant la dépendance vis-à-vis du biais la question est encore ouverte sur les arbres de Galton-Watson. Il paraitdonc un peu trop ambitieux d'attaquer directement le problème sur Zd. La diculté decette question est à mettre en relation avec les exemples surprenants de [69].

L'un des résultats obtenus lors de cette thèse concerne l'étude de la vitesse en fonctiondu paramètre de percolation. Il est naturel de penser que la percolation crée des pièges,des culs-de-sac dans la direction de la dérive et diminue le nombre manières de partir àl'inni. Ainsi, intuitivement, eectuer une percolation ne devrait que pouvoir diminuer lavitesse de la marche. Pour tenter de répondre partiellement à cette question, nous avonscalculé la dérivée de la vitesse au point p = 1 vis-à-vis du paramètre de percolation etnous avons montré que dans un large spectre de cas, la marche est eectivement ralentie.

Pour être plus précis, introduisons les fonctions de Green de la conguration ω

pour tout x, y ∈ Zd, Gω(x, y) := Eωx

[∑n≥0

1Xn ∈ y].

Rappelons tout d'abord que v`(1) =∑

e∈ν pω0(0, e)e, où ω0 est l'environnement à

p = 1, i.e. s'il n'y a pas eu percolation. De plus nous posons p(e) = pω0(0, e) et ν lesvecteurs unités de Zd.

Théorème 1.1. Pour d ≥ 2, p ∈ (pc(d), 1) et pour tout ` ∈ Rd∗, on a

v`(1− ε) = v`(1)− ε∑e∈ν

(v`(1) · e)(Gωe0(0, 0)−Gωe0(e, 0))(v`(1)− de) + o(ε),

où pour tout e ∈ ν on note

pour f ∈ E(Zd), ωe0(f) = 1f 6= e and de =∑e′∈ν

pωe0(0, e′)e′,

respectivement l'environnement où seule l'arête [0, e] est fermée et la dérive correspon-dante en 0.

26

1. COMPORTEMENT DE LA VITESSE SUR LE CLUSTER DEPERCOLATION VIS-À-VIS DES PARAMÈTRES

Proposition 1.1. Notons Je = Gω0(0, 0) − Gω0(e, 0) pour e ∈ ν. On peut réécrire leterme d'ordre 1 dans le développement précédent de la manière suivante

v′`(1) =∑e∈ν

(v`(1) · e) p(e)Je

1− p(e)Je − p(−e)J−e (e− v`(1)),

où les fonctions de Green intervenant ne dépendent que de l'environnement ω0. On peutainsi montrer que si pour tout e ∈ ν tel que v`(1) · e > 0 on a v`(1) · e ≥ ||v`(1)||22 alors

v`(1) · v′`(1) > 0,

ce qui montre que la percolation ralentit la marche au moins à p = 1.La condition précédente est vériée dans les deux cas suivants

1. ~ ∈ ν, i.e. si la dérive est suivant un des axes,

2. ` = λ~, où λ < λc(~) pour un certain λc(~) > 0, i.e. quand la dérive de la marcheest susament faible.

Remarque 1.1. La propriété de monotonie de la Proposition 1.1 devrait être vraie pourtoutes dérives, mais pour des raisons techniques nous n'arrivons pas à faire aboutir lescalculs. Plus généralement on s'attend à ce que cette propriété soit vraie dans une largegamme de cas, par exemple dans tout le régime sur-critique. Pour une conjecture reliéeà ces phénomènes, voir [18].

Remarque 1.2. Une conséquence non triviale du théorème est que la vitesse est locale-ment non nulle autour de p = 1.

Remarque 1.3. Finalement ce résultat nous donne quelques idées quant à la dépendencede la vitesse vis-à-vis du biais. En eet, xons un biais ` et un certain µ > 1, alors leThéorème 1.1 implique que pour tout ε0 = ε0(`, µ) > 0 susament petit, on a

vµ`(1− ε) · ~ > v`(1− ε) · ~ pour ε < ε0.

La démonstration de ce résultat s'inspire d'un résultat de Sabot [90] qui étudie unenvironemment invariant par translation déterministe soumis à une petit modicationaléatoire et i.i.d. en chacun de ses sites. Ici le contexte est assez diérent car notre en-vironnement est soumis à une perturbation très forte, on perd l'ellipticité de la marche,mais rare. On doit en quelque sorte démontrer que les eets potentiellement non-bornésqui proviennent d'une arête enlevée, sont petits une fois que l'on a moyenné sur l'envi-ronnement.

L'outil central pour la démonstration de ce théorème est la fonction de Green, quid'une part est reliée à la vitesse et d'autre part est un outil que nous savons étudier viales théorèmes de Kalikow.

27


Voici une rapide esquisse de la preuve de la continuité de la vitesse, les outils pourobtenir la dérivée étant essentiellement similaires. On note pour x, y ∈ Zd, P un opéra-teur Markovien et δ < 1, la fonction de Green tuée géométriquement avec un paramètre1− δ

GPδ (x, y) := EP

x

[ ∞∑k=0

δk1Xk = y]and Gω

δ (x, y) := GPω

δ (x, y),

où P ω est l'opérateur Markovien associé à la marche dans l'environnement ω.On introduit alors l'environnement de Kalikow associé au point 0 et à l'environ-

nement P1−ε[ · | I ], qui est donné pour z ∈ Zd, δ < 1 et e ∈ ν par

pεδ(z, e) =E1−ε[G

ωδ (0, z)pω(z, e)|I]

E1−ε[Gωδ (0, z)|I]

.

L'un des résulats démontrés par Kalikow [55], se généralise facilement de la manièresuivante

Proposition 1.2. Pour z ∈ Zd et δ < 1, on a

E1−ε

[Gωδ (0, z)|I

]= G

bpεδδ (0, z).

On peut directement adapter la preuve de la Proposition 1 de [90], qui ne nécessitepas d'hypothèse d'uniforme ellipticité dans le cas δ < 1.

Le lien entre les fonctions de Green et la vitesse asymptotique vient de la propositionsuivante

Proposition 1.3. Pour tout 0 < ε < 1− pc(Zd), on a

limδ→1

∑z∈Zd G

bωεδδ (0, z)dεδ(z)∑

z∈Zd Gbωεδδ (0, z)

= limδ→1

E[Xτδ ]

E[τδ]= v`(1− ε),

où dεδ(z) =∑e∈ν

pεδ(z, e)e.

En notant Cεδ l'enveloppe convexe des d

εδ(z) pour z ∈ Zd, une conséquence immédiate

de la proposition précédente est que

Proposition 1.4. Pour ε > 0 on a que v`(1 − ε) est un point d'accumulation de Cεδ

quand δ tend vers 1.

Ces deux propositions sont démontrées dans la preuve de la Proposition 2 de [90] etreposent uniquement sur l'existence d'une vitesse asymptotique, ce qui est vérié par lerésultat de [99]. Introduisons alors les notations

I = il existe un unique cluster inni passant par 0,

28

2. UN LIEN ENTRE LES M.A.M.A. ET UN MODÈLE DE PIÈGEJOUET

etC(z) = e ∈ ν, [z, z + e] est fermée,

où C(z) désigne donc l'ensemble des arêtes adjacentes à z qui sont fermées dans lapercolation.

En omettant l'indice ε dans E1−ε[ · ], on peut alors comprendre dεδ(z) en décomposantla dérive de Kalikow suivant les congurations en z

dεδ(z) =∑e∈ν

∑A⊂ν

E[1I1C(z) = AGω

δ (0, z)p(z, e)e]

E[1IGω

δ (0, z)] (1.1)

=∑

A⊂ν, A6=ν

E[1I1C(z) = AGω

δ (0, z)]

E[1IGω

δ (0, z)] dA

=∑

A⊂ν, A6=ν

P[C(z) = A]E[1IGω

δ (0, z)|C(z) = A]

E[1IGω

δ (0, z)] dA,

où dA =∑e∈A

pA(e)e est la dérive sous la conguration A. Ici pA(·) désigne donc pωA0 (0, ·)

où ωA0 est l'environnement où toutes les arêtes sont ouvertes sauf les arêtes adjacentesà 0 dans ν \ A qui sont fermées.

Il est alors susant de montrer qu'uniformément en z et en A on peut avoir

E[1IGω

δ (0, z)|C(z) = A]

E[1IGω

δ (0, z)] < C,

ce qui entraine quedεδ(z) = d∅ +O(ε) = v`(1) +O(ε),

ce qui permet d'obtenir appliquer la Proposition 1.4 et obtenir la continuité de la vitesse.Cette estimée technique se révèle dicile à traiter et constitue une grosse partie de

l'article.

2 Un lien entre les M.A.M.A. et un modèle de piège

jouet

Le résultat principal obtenu dans [8] concerne la limite d'échelle de la marche aléa-toire biaisée sur un Galton-Watson avec des feuilles.

29


Théorème 2.1. Notons, γ = − ln f ′(q)/ ln β. Pour tout λ > 0, en notant nλ(k) =bλf ′(q)−kc, on a

∆nλ(k)

nλ(k)1/γ

d−→ Yλ

où Yλ a une loi inniment divisible. Ce qui implique que

ln |Xn|lnn

→ γ.

Cependant pour β susament grand, la suite (∆n/n1/γ)n≥0 ne converge pas en loi.

Cette section est dédiée à une explication de l'intuition qui se situe derrière la preuvede ce résultat. Nous cherchons à présenter les grandes lignes de la démonstration carnous pensons que la méthode d'analyse est susament robuste pour s'étendre à plusieursautres modèles.

Un point central des travaux eectués dans [35], [36], [34], [8] and [9] est la clarica-tion des liens entre les M.A.M.A. qui sont réversibles, transientes dans une direction età vitesse nulle avec un modèle de piège jouet.

Il est clair depuis longtemps que le ralentissement, qui provoque un régime sous-balistique, est essentiellement lié à l'existence de pièges dans l'environnement danslesquels la marche demeure susament longtemps. Cependant ce n'est que très récem-ment qu'une méthode d'analyse précise de ses modèles a commencé à prendre forme.La méthode n'est pas encore complète au sens où dans un cadre général nous ne savonspas comment dénir la notion de piège et que l'analyse du temps passé dans un piègereste à faire au cas par cas, mais des similarités apparaissent. Le but de cette analyseest d'obtenir des théorèmes de convergence du type

il existe γ < 1,Xn

nγloi−→ L.

Nous allons essayer de donner les grandes étapes de la preuve type, en illustrant viadeux modèles concrets qui sont aujourd'hui bien compris

1. la M.A.M.A. uni-dimensionnelle dans le régime transient vers l'inni à vitessenulle,

2. la marche aléatoire biaisée sur un Galton-Watson avec des feuilles (p0 > 0) dansle régime sous-balistique.

Nous commençons par nous intéresser au temps d'atteinte du niveau n dans la di-rection de la transience directionelle qu'on l'on peut relié à Xn via un argument d'in-version classique. On pose donc ∆n = infi ≥ 0, |Xi| = n dans le cas de l'arbre et∆n = infi ≥ 0, Xi = n sur Z.

30


Tout d'abord nous avons besoin de l'existence d'une structure de régénération, cequi nous assure que le nombre de sites vus pour atteindre le niveau n que l'on noteraSn vérie

Sn ∼ CSn,

pour un certain CS > 0. Cette propriété est naturelle, en eet dans un bloc de régénéra-tion la marche avance d'un nombre de pas d'espérance ni et voit un nombre ni desites.

Lorsque la marche parcourt l'environnement elle rencontre des pièges qui sont àl'origine de son ralentissement. Pour simplier nous allons nous imaginer qu'en chaquesite notre marche rencontre un piège. Ainsi à chaque fois qu'on se situe sur un siteparticulier nous avons un temps d'attente dépendant du piège (qui est aléatoire) présenten ce site.

Vu de loin, la marche est essentiellement uni-dimensionnelle par transience direction-nelle. Dans un souci de simplication nous allons supposer dans la suite que la marcheest sur Z. De plus nous allons supposer que la marche va toujours d'un piège au suivant,cette hypothèse est à première vue abusive, mais nous allons la justier à postériori.

Finalement, nous avons donc assimilé nos deux modèles au modèle simplié deM.A.M.A. sur N suivant : Xt = i pour

∑ij=1 T

(j)tot ≤ t <

∑i+1j=1 T

(j)tot , où T

(j)tot est le temps

total passé dans le j-ème piège vu. On peut espérer que cela soit une bonne représen-tation pour des modèles assez généraux de transience directionelle à vitesse nulle. Nousreprésentons dans la gure 3.1 ce modèle simplié, où les pièges sont en pointillé.

1 2 3 4 5 6 7 8

T(1)tot T

(3)tot T

(5)tot T

(8)tot

Fig. 3.1 Modèle de piège simplié

Après ces simplications notre problème devient plus abordables en eet on remarqueque ∆n correspond essentiellement au temps passé dans les Csn premiers pièges et donc

∆n ≈CSn∑i=1

T(i)tot,

est une somme de v.a.i.i.d..

31


Dans le cas de la M.A.M.A. uni-dimensionnelle, ces pièges sont caractérisés commedes puits de potentiel. Dans le régime transient vers ∞, ces puits sont caractérisés parde grandes montées du potentiel (voir gure 3.2).

x

V (x)

0

potentiel

puit de potentiel

Fig. 3.2 Puit de potentiel dans la régime transient vers ∞ sur Z

Dans le cas des marches aléatoires biaisées sur un Galton-Watson avec feuilles, ils'agit des feuilles dans le sens où on considère l'ensemble des sommets qui possèdentune descendance nie (voir Figure 3.3).

La tâche dicile est d'identier les pièges et d'être capable d'en décrire la structure.Dans ces deux modèles, on remarque que pour partir à l'inni à partir d'un piège, lamarche doit nécessairement passer par des zones où la mesure invariante est beaucoupplus faible. Ceci est probablement une piste assez générale pour identier les pièges.

En eet, supposons qu'à partir d'un point x la marche doit passer par un point yde mesure invariante inférieure. On peut alors, via un argument de réversibilité, obtenirune majoration de la probabilité de sortie d'un piège car

P xω [T+

x <∞] ≤ P xω [T+

x < Ty] ≤ π(y)

π(x),

ce qui montre déjà que typiquement la marche va passer un temps de l'ordre de π(x)/π(y)en x.

Dans le cadre de Z, on obtient donc une majoration de la probabilité de sortie d'unpiège en e−H où H est l'augmentation du potentiel dans le piège, i.e.

H = maxx∈piège

[max

y≥x,y∈piègeV (y)− V (x)

].

32


tronc

feuille

∞ ∞Fig. 3.3 Les feuilles dans un Galton-Watson

Dans le cadre de la marche aléatoire biaisée sur le Galton-Watson avec des feuilles,la quantité qui nous intéresse est la hauteur du piège (qui est directement reliée à lamesure invariante) H, i.e. le nombre de niveaux distincts dans le piège, qui nous permetd'obtenir une majoration du temps de sortie du piège de l'ordre de β−H .

Analysons le comportement de la marche dans un grand piège, sous l'hypothèsesimplicatrice que ces deux majorations sont en fait des égalités. Deux points vontjouer un rôle important cette analyse,

1. tout d'abord il y a le fond du piège, i.e. le point de mesure invariante maximale,que le notera δ,

2. ensuite il y a la sortie s (dans nos deux modèles on peut identier cette sortie à unpoint) qui est un ensemble du piège à partir duquel il est facile de partir à l'inni.

Ce point de sortie est choisi, dans le cas de la marche aléatoire sur Z comme le pointqui est au sommet de la vallée du côté droit et dans le cas de la M.A.M.A. sur l'arbrecomme le point du tronc auquel la feuille est attachée.

Le phénomène qui se produit est que la marche va eectuer un grand nombre d'ex-cursions à partir du fond du piège avant d'atteindre la sortie. Ce nombre d'excursionsest une géométrique de paramètre p = Pω[T+

δ < Ts | X0 = δ] et on peut alors approximerle temps passé durant un passage dans le piège par

Tpiège =

Geom(p)∑i=1

T (i)exc ≈ Geom(p)Eω[T (i)

exc] ≈1

pEω[T (i)

exc]e,

où e désigne une exponentielle de paramètre 1 qui ne dépend que de la marche et les T (i)exc

sont des variables i.i.d. distribuées comme le temps d'une excursion à partir de δ qui ne

33


sort pas du piège. Ici nous avons utilisé une sorte de loi des grands nombres associée àune approximation d'une géométrique par une exponentielle. Elles sont toutes les deuxjustiées par le fait que le comportement de la marche est essentiellement déterminé parce qui se passe dans les grands pièges, i.e. dans le cas où p est petit, qui nous permetd'obtenir les approximations faites au-dessus.

Ensuite il est possible que la marche ayant atteint s puisse retourner au fond decelui-ci. Nous introduisons

W = cardi ∈ N | Xi = s, ∃j ≥ i, Xj = δ, Xk 6= s, ∀i < k < j,le nombre d'entrées profondes dans le piège. Le temps total passé dans un grand piègeest donc

Ttot =1

pEω[T (1)

exc]W∑i=0

ei := Z∞1

p, (2.1)

où les ei sont des v.a.i.i.d. indépendantes de loi exponentielle de paramètre 1. La variablealéatoire W est essentiellement indépendantes des autres variables aléatoires car elledépend surtout de ce qui se passe à l'extérieur du piège.

Sans entrer trop dans les détails, la formule précédente signie que Ttot est essen-tiellement déterminé par 1/p car il est possible de montrer que E[Z∞] <∞.

Dans les cas que nous développons ici, nous rappelons que 1/p ≈ eH (resp. 1/p ≈ βH)dans le cas de la M.A.M.A. uni-dimensionnelle (resp. marche aléatoire biaisée sur unGalton-Watson avec des feuilles). Dans les deux cas, il parait alors naturel de classerles pièges en fonction de leur impact qui est lié à leur taille qui est quantié via lavariable aléatoire H.

Il nous reste donc à déterminer l'ordre de grandeur de la queue de ces variablesaléatoires. Dans le cas uni-dimensionnel, sous l'hypothèse que ln ρ0 a une distributionqui n'est pas concentrée sur un réseau, on a un résultat de Iglehart [54] qui nous permetd'obtenir que

P[H ≥ t] ∼ CIe−κt i.e. P[eH ≥ t] ∼ CIt

−κ,

où κ est tel que E[ρκ0 ] = 1.Dans le cadre de la marche sur l'arbre, nous utilisons un résultat de Heathcote, Seneta

et Vere-Jones [50] qui nous permet de dire que

P[H ≥ n] ∼ αf ′(q)n i.e. P[eH ≥ t] = Θ(t−γ),

où f(z) =∑

k≥0 pkzk, q désigne son unique point xe dans (0, 1) et γ = − ln f ′(q)/ ln β.

Nous n'avons pas ici à proprement parler d'équivalent. En eet H est à valeurs entières,ceci qui correspond à ce que ln ρ0 soit concentrée sur un réseau en dimension 1.

En se remémorant (2.1), nous voyons dans le cas uni-dimensionnel

P[Ttot ≥ t] =

∫P[1

pu ≥ t

]dP[Z∞ ∈ du] ∼

∫CIt

−κuκdP[Z∞ ∈ du] = CIE[Zκ∞]t−κ,

34


ce qui nous d'appliquer des théorèmes classiques sur les variables aléatoires à queueslourdes et à variations régulières, voir [31], pour montrer un théorème de convergencevers une loi stable. En notant T (i)

tot une suite de v.a.i.i.d. de loi Ttot, on a

∆n

n1/κ=

∑CSni=1 T

(i)tot

n1/κ→ Sκ,

où Sκ est une loi stable complétement asymétrique d'indexe κ, dont on peut calculer lesautres paramètres de manière explicite en fonction de certain moments de la variableZ∞ et de CS.

Nous avons ici omis deux points importants,

1. les temps passés dans diérents pièges ne sont pas indépendants,

2. la marche ne passe pas d'un piège au suivant sans jamais revenir en arrière.

Ces propriétés sont à proprement parler fausses, mais si nous regardons des événe-ments génériques, i.e. des convergences en loi par opposition à des grandes déviations,ces deux propriétés sont essentiellement vériées. En eet, les sommes de v.a.i.i.d. àqueues lourdes sont concentrées sur les plus gros termes, par exemple sur une somme den la somme est concentrée sur les nε plus gros termes, elles correspondent donc aux plusgrosses vallées qui sont génériquement à grande distance. Ainsi le nombre de retoursdans chacunes des grosses vallées sont essentiellement indépendants ce qui nous donnela première propriété. De plus une fois qu'une vallée est atteinte il est peu vraisemblablede revenir contre la direction de la transience entre deux grands pièges, i.e. sur unedistance polynomiale. Cela fournit la deuxième propriété.

Un raisonnement similaire peut-être fait dans le cas de l'arbre. Seulement dans ce casil n'y pas de possibilité de convergence en loi. En eet le temps d'atteinte du niveau n estréduit à l'étude de la convergence d'une suite de v.a.i.i.d. qui ont des queues qui ne sontpas à variations régulières et par des résultats généraux sur les tableaux triangulaires,voir [80], on ne peut pas trouver de renormalisation convenable. Cependant grâce àl'application d'un résultat de [80] (que l'on retrouvera reformulé dans le Théorème 10.6),on peut obtenir une convergence selon des sous-suites

pour λ ∈ [1, β),∆(λβ)γk

(λβ)k=

1

βk

bβkγc∑i=1

T(i)tot

d−→ Y,

où Y est une variable aléatoire de loi inniment divisible qui possède une partie brown-ienne nulle.

Ce raisonnement explique aussi essentiellement les résultats obtenus dans [35] et [36].Il est le coeur de la démonstration dans [8] pour traiter le cas des marches biaisées surun Galton-Watson avec des feuilles dont le résultat a été cité au début de cette section.

Plus généralement les résultats qu'on peut obtenir sont reliés au Bouchaud's TrapModel qui fut introduit dans [17]. Nous ne voulons pas introduire également ce modèle

35


et nous renvoyons le lecteur au mini-cours [7] pour une introduction générale. Noussignalons également l'article [107] qui est encore plus fortement relié aux modèles quenous avons considéré ici. En eet le modèle de Bouchaud dirigé qui est étudié danscet article possède des propriétés similaires à celles des M.A.M.A. présentées ici mêmeconcernant les événements rares du types grandes déviations.

3 Déviations modérées pour la M.A.M.A. sur ZNous venons de voir dans la section précédente que la M.A.M.A. uni-dimensionnelle

était fortement lié à un modèle jouet, du moins dans le cas d'un comportement typiquede la marche. Cependant lorsqu'il s'agit de regarder des événements rares certainesapproximations faites dans la section précédente sont un peu abusives. En particulieril n'est pas vrai que la particule avance essentiellement d'un piège à un autre, il est eneet possible d'observer de grands retours en arrière. Une modélisation plus adéquate dumodèle serait alors de conserver un temps d'attente à chaque site du type eHe où e unevariable aléatoire exponentielle de paramètre 1 et H est une variable aléatoire distribuéecomme la hauteur d'une vallée, mais de remplacer les sauts de la marche aléatoire quiétaient systématiquement vers la droite par une marche aléatoire biaisée vers la droite.

Cette représentation simpliée nous permet d'aborder des problèmes de dévia-tions modérées. Sous les hypothèses du théorème de Kesten, Kozlov et Spitzer, i.e.si E[ln ρ0] < 0 et E[ρκ0 ] = 1, nous avons que

limn→∞

lnXn/ lnn = κ, P-p.s..

Une question naturelle à considérer est de savoir la probabilité d'un écart par rapportà cet événement. Nous cherchons à comprendre les événements rares suivant

le ralentissement, ce qui signie qu'au temps n la particule est à gauche de nν0

où ν0 < 1 ∧ κ, ce qui signie que la particule est beaucoup plus lente que soncomportement typique,

le recul, ce qui signie qu'au temps n la particule est à gauche de −nν , l'accélération, ce qui signie que la particule est à droite de nν0 avec κ < ν0 < 1.Nous désignons tous ces événements par des déviations modérées. En eet, dans

le cas de ralentissement, la fonction de taux pour les grandes déviations I(·) vérieque I(0) = 0, ce qui nous permet seulement de dire que les probabilités décroissent demanière sous-exponentielle en n (voir [20]). Notre but est alors de préciser ce résultaten identiant le coecient de décroissance sous-exponentielle.

Nous obtenons plusieurs résultats, en particulier des déviations, quenched ou an-nealed, pour la marche ou pour le temps d'atteinte, dans le cas uni-dimensionnel nor-mal ou bien rééchi en 0. L'ensemble de ces résultats est assez longs à énoncer et nousrenvoyons le lecteur à la Section 6 pour tous les énoncés.

36

3. DÉVIATIONS MODÉRÉES POUR LA M.A.M.A. SUR Z

On peut résumer une partie des résultats sous la forme d'un graphique. Dans le cassous-balistique, i.e. κ ∈ (0, 1), les déviations modérées quenched pour la marche Z sontreprésentées en 3.4 via le graphique de la fonction :

f(ν) =

limn→∞ ln(− lnPω[Xn < −n−ν ])/ lnn, si ν ∈ (−1, 0],limn→∞ ln(− lnPω[Xn < nν ])/ lnn, si ν ∈ (0, κ),limn→∞ ln(− lnPω[Xn > nν ])/ lnn, si ν ∈ [κ, 1).

−1 − κκ+1

0 κκ+1

κ 1

1f(ν)

ν

κκ+1

Fig. 3.4 Graphique de f(ν), −1 < ν < 1

Les résultats obtenus dans le cas du ralentissement, qui constitue le cas le plusdicile, sont en fait les mêmes que ce que l'on obtient en faisant les calculs dans lemodèle simplié présenté dans le début de cette section. Il faut analyser la compétitionqu'il y a entre les deux possibilités suivantes

1. aller chercher de grandes vallées dans lesquelles il est naturel de passer beaucoupde temps,

2. passer un temps anormalement grand dans une vallée normale.

Une ébauche détaillée de preuve se trouve dans l'article [39].

37


38

4Biased random walks on

Galton-Watson trees with leaves

We consider a biased random walk Xn on a Galton-Watson tree with leaves in thesub-ballistic regime. We prove that there exists an explicit constant γ = γ(β) ∈ (0, 1),depending on the bias β, such that Xn is of order nγ. Denoting ∆n the hitting timeof level n, we prove that ∆n/n

1/γ is tight. Moreover we show that ∆n/n1/γ does not

converge in law (at least for large values of β). We prove that along the sequencesnλ(k) = bλβγkc, ∆n/n

1/γ converges to certain innitely divisible laws. Key tools forthe proof are the classical Harris decomposition for Galton-Watson trees, a new variantof regeneration times and the careful analysis of triangular arrays of i.i.d. heavy-tailedrandom variables.

The material of this chapter is a joint work with G. Ben Arous, N. Gantert and A.Hammond and has been submitted for publication, see [8].

CHAPITRE 4. BIASED RANDOM WALKS ON GALTON-WATSONTREES WITH LEAVES

1 Introduction and statement of the results

Consider a supercritical Galton-Watson branching process with generating functionf(z) =

∑k≥0 pkz

k, i.e. the ospring of all individuals are i.i.d. copies of Z, where P[Z =k] = pk. We assume that the tree is supercritical and has leaves, i.e. m := E[Z] =f ′(1) ∈ (1,∞) and p0 > 0. We denote by q ∈ (0, 1) the extinction probability, whichis characterized by f(q) = q. Starting from a single progenitor called root and denotedby 0, this process yields a random tree T . We will always condition on the event ofnon-extinction, so that T is an innite random tree. We denote (Ω,P) the associatedprobability space : P is the law of the original tree, conditioned on non-extinction.

For ω ∈ Ω, on the innite Galton-Watson tree T (ω), we consider the βbiasedrandom walk as in [68]. More precisely, we dene, for β > 1, a Markov chain (Xn)n∈Non the vertices of T , such that if u 6= 0 and u has k children v1, . . . , vk and parent ←−u ,then

1. P [Xn+1 =←−u |Xn = u] = 11+βk

,

2. P [Xn+1 = vi|Xn = u] = β1+βk

, for 1 ≤ i ≤ k,

and from 0 all transitions to its children are equally likely. This is a reversible Markovchain, and as such, can be described as an electrical network with conductances c(←−x , x) :=β|x|−1 on every edge of the tree (see [70] for background on electrical networks).

We always take X0 = 0, that is we start the walk from the root of the tree. We denoteby P ω[·] the law of (Xn)n=0,1,2,... and we dene the averaged law as the semidirect productP = P× P ω.

Many interesting facts are known about this walk (see [68]). As one might expect, itis transient. Denote by |u| = d(0, u) the distance of u to the root. It is known that P-a.s.,|Xn|/n converges to a deterministic limit v. Moreover, the random walk is ballistic, i.e.its limiting velocity v > 0, if and only if β < βc = 1/f ′(q). In the subballistic regime, i.e.if β ≥ βc, we have v = 0. The reason for the subballistic regime is that the walk losestime in traps of the tree, from where it cannot go to innity without having to go for along time against the drift which keeps it into the trap. The hypothesis p0 > 0 is crucialfor this to happen.

As in all subballistic models, a natural question comes up : what is the typicaldistance of the walker from the root after n steps ? This is the question we address inthis paper. We always assume that

E[Z2] <∞and

β > 1/f ′(q) ,

recalling that 1/f ′(q) > 1. We introduce the exponent

γ :=− ln f ′(q)

ln β=

ln βcln β

< 1 (1.1)

40

1. INTRODUCTION AND STATEMENT OF THE RESULTS

so that βγ = 1/f ′(q).Let ∆n be the hitting time of the n-th level :

∆n = infi ≥ 0, |Xi| = n.

Theorem 1.1. (i) The laws of (∆n/n1/γ)n≥0 under P are tight.

(ii) The laws of (|Xn|/nγ)n≥0 under P are tight.(iii) We have

limn→∞

ln |Xn|lnn

= γ, P− a.s.. (1.2)

Of course, this raises the question of convergence in distribution of the sequence(∆n/n

1/γ)n≥0. The next theorem gives a negative answer.

Theorem 1.2. For β large enough, the sequence (∆n/n1/γ)n≥0 does not converge in

distribution.

However, we can establish convergence in distribution along certain subsequences.

Theorem 1.3. For any λ > 0, denoting nλ(k) = bλf ′(q)−kc, we have

∆nλ(k)

nλ(k)1/γ

d−→ Yλ

where the random variable Yλ has an innitely divisible law µλ.

We now describe the limit laws µλ. For some constants ρ and Ca (the constant ρ isdened in (2.2), the constant Ca in Lemma 6.1), we have

Yλ = (ρCaλ)1/γ Y(ρCaλ)1/γ

whereYλ has the law I(dλ, 0,Lλ) .

The innitely divisible law I(dλ, 0,Lλ) is given by its Lévy representation (see [80], p.32). More precisely, the characteristic function of I(dλ, 0,Lλ) can be written in the form

E[eit

eYλ] =

∫eitxI(dλ, 0,Lλ)(dx) = exp

idλt+

∞∫0

(eitx − 1− itx

1 + x2

)dLλ(x)

where dλ is a real constant and Lλ a real function which is non-decreasing on the

interval (0,∞) and satises Lλ(x) → 0 for x → ∞ anda∫0

x2dLλ(x) < ∞ for every

a > 0. Comparing to the general representation formula in [80], p. 32, we here have that

41


the gaussian part vanishes and Lλ(x) = 0 for x < 0. The function Lλ is called the Lévyspectral function. Note that Lλ is not a Lévy-Khintchine spectral function.

In order to describe Lλ, dene the random variable

Z∞ =S∞

1− β−1

Bin(W∞,p∞)∑i=1

ei, (1.3)

where p∞ = 1− β−1 is the escape probability of a β-biased random walk on N. Further,the random variables ei in (1.3) are i.i.d. exponential random variables of parameter 1and the non-negative random variables (ei), W∞ and S∞ in (1.3) are independent. Therandom variables S∞ andW∞ will be described in (3.6) and Proposition 6.1 respectively.The random variable Bin(W∞, p∞) is of law Binomial with parametersW∞ and p∞. Now,denoting by F∞(x) = P[Z∞ > x] the tail function of Z∞, we have

Theorem 1.4. (i) The Lévy spectral function Lλ is given by

L1(x) =

0 if x < 0,

−(1− β−γ)∑k∈Z

βγkF∞(xβk) if x > 0.

(ii) for all λ ∈ [1, β) and x ∈ R, Lλ(x) = λγL1(λx) and Lβ(x) = L1(x).(iii) dλ is given by

dλ = λ1+γ(1− β−γ)∑k∈Z

β(1+γ)kE[ Z∞

(λβk)2 + Z2∞

].

(iv) Lλ is absolutely continuous.(v) The following bounds hold

1

βγE[Zγ∞]

1

xγ≤ −L1(x) ≤ E[Zγ∞]

1

xγ. (1.4)

(vi) The measure µλ is absolutely continuous with respect to Lebesgue measure and hasa moment of order α if and only if α < γ.(vii) When β is large enough, xγLλ(x) is not a constant.(viii) The random variable Z∞ has an atom at 0 and a smooth density ψ on (0,∞).Further, Z∞ has nite expectation.

Remark 1.1. We believe that Theorem 1.2 holds true for all values β > βc. The proofwould amount to showing that the function xγL1(x), with L1(x) given in Theorem 1.4,is not a constant.

42

1. INTRODUCTION AND STATEMENT OF THE RESULTS

Next we explain briey, using a toy example, the reason for the non-convergenceof (∆n/n

1/γ)n≥0 and the convergence of subsequences in Theorems 1.2 and 1.3. Thereasons lie in the classical theory of sums of i.i.d. random variables. Consider a sequenceof i.i.d. random variables Gi, geometrically distributed with parameter a. Let

Sn =n∑i=1

βGi .

It is easy to see, using classical results about triangular arrays of i.i.d. random variables(c.f. [80]), that for α = − log(1−a)

log β, and nλ(k) = β−αk, the distributions of

1

nλ(k)1/αSnλ(k) converge to an innitely divisible law

(see Theorem 10.1 for a more general result). But obviously here Sn/n1/α cannot con-verge in law, if α < 2, because one easily checks that the distribution of βG1 does notbelong to the domain of attraction of any stable law. This is the basis of our belief thatTheorem 1.2 should be valid for any β > βc.

We now discuss the motivation for this work. If one considers a biased random walkon a supercritical percolation cluster on Zd, it is known that, at low bias, the randomwalk is ballistic (i.e. has a positive velocity) and has gaussian uctuations, see [99] and[12]. It is also known that, at strong bias, the random walk is subballistic (i.e. the velocityvanishes). It should be noted that, in contrast to the Galton-Watson tree, the existence ofa critical value separating the two regimes is not established for supercritical percolationclusters. The behaviour of the (law of) the random walk in the subballistic regime is avery interesting open problem. It was noted in [101] that the behaviour of the randomwalk in this regime is reminiscent of trap models introduced by Bouchaud (see [17] and[7]). Our work indeed substantiates this analogy in the simpler case of supercriticalrandom trees. We show that most of the time spent by the random walk before reachinglevel n is spent in deep traps. These trapping times are roughly independent and areheavy-tailed. However, their distribution does not belong to the domain of attraction ofa stable law, which explains the non-convergence result in Theorem 1.2.

We note that it is possible to obtain convergence results to stable laws if one getsrid of the inherent lattice structure. One way to do this is to randomize the bias β. Thisis the approach of the forthcoming paper [9].

For other recent interesting works about random walks on trees, we refer to [53],[2], and [78]) .

There is also an analogy with the one-dimensional random walk in an i.i.d. randomenvironment (RWRE). This model also shows a ballistic and a subballistic regime, ex-plicitly known in terms of the parameters of the model. We refer to [104] for a survey.In the subballistic regime, it was shown in [59] that depending on a certain parameterκ ∈ (0, 1], and under a non-lattice assumption, Xn

nκconverges to a functional of a stable

43


law, if κ < 1, and Xnn/ lnn

converges to a functional of a stable law, if κ = 1. Recently,using a precise description of the environment, [36] and [35] rened this last theorem bydescribing all the parameters of the stable law, in the case κ < 1.

Our method has some similarity to the one used in [35]. In comparison to [35], anadditional diculty arises from the fact the traps met depend not only on the environ-ment but also on the walk. Moreover one has to take into account the number of timesthe walker enters a trap, which is a complicated matter because of the inhomogeneity ofthe tree. This major technical diculty can be overcome by decomposing the tree andthe walk into independent parts, which we do using a new variant of regeneration times.

The paper is organized as follows : In Section 2 and Section 3 we explain how todecompose the tree and the walk. In Section 4 we give a sketch of the proof of Theorem1.3. Sections 5 - 9 prepare the proof of Theorem 1.3 and explain why the hitting timeof level n is comparable to a sum i.i.d. random variables. Section 10 is self-containedand its main result, Theorem 10.1, is a classical limit theorem for sums of i.i.d. randomvariables which is tailored for our situation. In Section 11, we nally give the proofs ofthe results. In Subsection 11.1, we apply Theorem 10.1 to prove Theorem 1.3. Subsection11.2 is devoted to the proof of Theorem 1.2, Subsection 11.3 gives the proof of Theorem1.1 and Subsection 11.4 the proof of Theorem 1.4.

Let us give some conventions about notations. The parameters β and (pk)k≥0 willremain xed so we will usually not point out that constants depend on them. Mostconstants will be denoted c or C and their value may change from line to line to easenotations. Specic constants will have a subscript as for example Ca. We will alwaysdenote by G(a) a geometric random variable of parameter a, with law given by P [Ga ≥k] = (1− a)k−1 for k ≥ 1.

2 Constructing the environment and the walk in the

appropriate way

In order to understand properly the way the walk is slowed down, we need to de-compose the tree. Set

g(s) =f((1− q)s+ q)− q

1− q and h(s) =f(qs)

q. (2.1)

It is known (see [65]), that a f -Galton-Watson tree (with p0 > 0) can be generatedby

(i) growing a g-Galton-Watson tree Tg called the backbone, where all vertices havean innite line of descent,

(ii) attaching on each vertex x of Tg a random number Nx of h-Galton-Watson trees,acting as traps in the environment T ,

44

2. CONSTRUCTING THE ENVIRONMENT AND THE WALK IN THEAPPROPRIATE WAY

where Nx has a distribution depending only on degTg(x) and given Tg and Nx the traps

are i.i.d., see [65] for details.

Fig. 4.1 The Galton-Watson tree is decomposed into the backbone (solid lines) andthe traps (dashed lines).

We will call bud a vertex at distance exactly one of the backbone. It is importantto consider the backbone together with the buds to understand the number of visits totraps.

It will be convenient to consider the attached Galton-Watson trees together withthe edge which connects them to the backbone. We dene a trap to be a graph (x ∪V, [x, y]∪E), where x is a vertex of the backbone, y is a bud adjacent to x and V (resp.E) are the vertices (resp. edges) of the descendants of y. The traps can themselves bedecomposed in a portion of Z called the spine, to which smaller trees called subtrapsare added, this construction is presented in detail in Section 3.

Let us now construct the random walk. We need to consider the walk on the backboneand on the buds, to this end we introduce

1. σ0 = σ′0 = 0,

2. σn+1 = infi > σn|Xi−1, Xi ∈ backbone,3. σ′n+1 = infi > σ′n|Xi−1, Xi ∈ backbone ∪ buds,

and we dene Yn = Xσn the embedded walk on the backbone, respectively Y ′n = Xσ′n

the embedded walk on the backbone and the buds.Moreover dene ∆Y

n = cardi ≥ 0|σi ≤ ∆n the time spent on the backbone to reachlevel n and similarly ∆Y ′

n = cardi ≥ 0|σ′i ≤ ∆n.

45


Denote, for a set A in the tree T+A = minn ≥ 1|Xn ∈ A, T+

y := T+y, TA :=

minn ≥ 0|Xn ∈ A, and Ty := Ty.Note that the process (Yn)n≥0 is a Markov chain on the backbone, which is inde-

pendent of the traps and the time spent in the traps. Here one has to be aware thatvisits to root do not count as time spent in a trap, precise denitions will follow below.Hence, in order to generate Yn we use a sequence of i.i.d. random variables Ui uniformlydistributed on [0, 1]. If Yj = w with Z∗1 children on the backbone, then

1. Yi+1 =←−w , if Ui ∈[0, 1

Z∗1β+1

],

2. Yi+1 = the jth-child of w, if Ui ∈[1− jβ

Z∗1β+1, 1− (j−1)β

Z∗1β+1

].

For background on regeneration times we refer to [102] or [104]. In the case of aβ-biased random walk Yn on Z, a time t is a regeneration time if

Yt > maxs<t

Ys and Yt < mins>t

Ys.

Denition 2.1. A time t is a super-regeneration time for Yn, if t is a regeneration timefor the corresponding β-biased random walk Yn on Z dened by

(i) Y0 = 0,

(ii) Yn+1 = Yn − 1, if Un ∈[0, 1

β+1

],

(iii) Yn+1 = Yn + 1 otherwise.

We denote t− SR the event that t is a super-regeneration time for Yn.It is obvious that a super-regeneration time for Yn is a regeneration time for Yn in

the usual sense (the converse is false).The walk can then be decomposed between the successive super-regeneration times(i) τ0 = 0,(ii) τi+1 = infj ≥ τi|j − SR.Since the regeneration times of a β-biased random walk on Z have some exponential

moments, there exists a > 1 such that E[aτ2−τ1 ] <∞ and E[aτ1 ] <∞.

Remark 2.1. The advantage of super-regeneration times compared to classical regen-eration times is that the presence of a super-regeneration time does not depend on theenvironment, but only the on the sequence (Ui)i≥0.

Remark 2.2. The drawback of super-regeneration times is that the event that k is asuper-regeneration time depends on the random variables (Ui)i≥0 and not only on thetrajectory of the random walk (Yn)n≥0.

Denoting for k ≥ 1, the σ-eld

Gk = σ(τ1, . . . , τk, (Yn∧τk)n≥0, x ∈ T (ω), x is not a descendant of Yτk).We have the following proposition

46

3. CONSTRUCTING A TRAP

Proposition 2.1. For k ≥ 1,

P[(Yτk+n − Yτk)n≥0 ∈ ·, x ∈ T (ω), x is a descendant of Yτk ∈ ·|Gk]=P[(Yn)n≥0 ∈ ·, T (ω) ∈ ·|0− SR].

Remark 2.3. The conditioning 0−SR refers only to the walk on the backbone, hence itis obvious that the behaviour of the walk in the traps and the number of times the walkerenters a trap is independent of that event.

We skip the proof of this Proposition for it is standard. A consequence of the propo-sition is that the environment and the walk can be subdivided into super-regenerationblocks which are i.i.d. (except for the rst one). As a consequence we have that

ρn :=cardY1, . . . , Y∆Y

n

nsatises ρn → ρ :=

E[cardYτ1 , . . . , Yτ2−1]E[τ2 − τ1]

,P− a.s (2.2)

which is the average number of vertices per level visited by Yn. This quantity is nitesince it is bounded above by than 1/v(β), where v(β) is the speed of |Yn| which is strictlypositive by a comparison to the β-biased random walk on Z.

When applying the previous proposition, it will be convenient to use the time-shiftfor the random walk, which we will denote by θ.

3 Constructing a trap

In the decomposition theorem for Galton-Watson trees, we attach to the verticesof the backbone a (random) number of h-Galton-Watson trees. We will denote theirdistribution with Q, hence Q[Z = k] = qk := pkq

k−1, where Z denotes the numberof children of a given vertex. As stated before the object we will denote a trap hasan additional edge : to describe a trap ` we take a vertex called root (or root(`) toemphasize the trap), link it to another vertex (denoted

−−→root(`)), which is the actual

root of a random h-Galton-Watson tree.When we use random variables associated to a trap, we refer to the random part of

that trap (the h-Galton-Watson tree). For example the notation Zn is the number ofchildren at the generation n with

−−→root being generation 0. In particular, we introduce

the height of a trapH = maxn ≥ 0, Zn > 0, (3.1)

and we say a trap has height k if H(`) = k, i.e. the distance between−−→root and the

bottom point of the trap is k.This way of denoting the random variables has the advantage that Zn (resp. H) are

distributed under Q, as the number of children at generation n (resp. the height) of ah-Galton-Watson tree.

47


The biggest traps seen up to level n are of size − lnn/ ln f ′(q), therefore a trap willbe considered big if its height is greater or equal to

hn =⌈(1− ε) lnn

− ln f ′(q)

⌉, (3.2)

for some ε > 0 which will eventually be chosen small enough. Such a trap will be calleda hn-trap or a big trap. It is in those traps that the walker will spent the majority ofhis time and therefore is important to have a good description of them.

The traps are (apart from the additional edge) subcritical Galton-Watson trees, assuch, they can be grown from the bottom following a procedure described in [43], thatwe recall for completeness. We will denote by δ the starting point of the procedure,corresponding to the leftmost bottom point of the trap, this last notation will be keptfor the whole paper.

With a slight abuse of notation, we will denote by Q a probability measure on anenlarged probability space containing the following additional information.

We denote by (φn+1, ψn+1) with n ≥ 0, a sequence of i.i.d pairs of random variableswith joint law given by

Q[φn+1 = j, ψn+1 = k] = cnqkQ[Zn = 0]j−1Q[Zn+1 = 0]k−j, 1 ≤ j ≤ k, k ≥ 1, (3.3)

where cn = Q[H=n]Q[H=n+1]

.Set T0 = δ. Construct Tn+1, n ≥ 0 inductively as follows :

1. let the rst generation size of Tn+1 be ψn+1,

2. let Tn be the subtree founded by the φn+1-th rst generation vertex of Tn+1,

3. attach independent h-Galton-Watson trees which are conditioned on having heightstricly less than n to the φn+1 − 1 siblings to the left of the distinguished rstgeneration vertex,

(iv) attach independent h-Galton-Watson trees which are conditioned on having heightstrictly less than n+ 1 to the ψn+1−φn+1 siblings to the right of the distinguishedrst generation vertex.

Then Tn+1 has the law of an h-Galton-Watson tree conditioned to have height n+ 1(see [43]).

We denote T the innite tree asymptotically obtained by this procedure ; from thistree we can obviously recover all Tn. If we pick independently the height H of a h-Galton-Watson tree and the innite tree T obtained by the previous algorithm, thenTH has the same law as a h-Galton-Watson tree.

We will call spine of this Galton-Watson tree the ancestors of δ. If y 6= δ is inthe spine, −→y denotes its only child in the spine. We dene a subtrap to be a graph(x ∪ V, [x, y] ∪ E), where x is a vertex of the spine, y is a descendant of x not on the

48


spine and V (resp. E) are the vertices (resp. edges) of the descendants of y. The vertexx is called the root of the subtrap, and we denote

Sx the set of all subtraps rooted at x . (3.4)

root

root

δ

Fig. 4.2 The trap is decomposed into the spine (solide lines) and the subtraps (dashedlines).

We denote by Si,j,kn+1 and Πi,j,kn+1 with n, i, j ≥ 0 and k = 1, 2, two sequences of inde-

pendent random variables, which are independent of (φn, ψn)n≥0 and given by

1. Sn+1,j,1n (resp. Sn,j,2n ) is the j-th subtrap conditioned to have height less than n

added on the left (resp. right) of the n+ 1-th (resp. n-th) ancestor of δ,

2. Πi,j,kn is the weight of Si,j,kn under the invariant measure associated to the conduc-

tances βi+1 between the level i and i+ 1, the root of Si,j,kn being counted as level0.

These random variables describe the subtraps and their weights.We denote Πi,j,k

−1 = 0 and

Λi(ω) =

φi−1∑j=1

Πi,j,1i−1 +

ψi−φi∑j=1

Πi,j,2i , (3.5)

49


which is the weight of the subtraps added to the i−th ancestor of δ.Due to the following lemma, the random variables Λi will be important to describe

the time spent in traps.

Lemma 3.1. Let (G, c(e)) be a nite or positive recurrent electrical network, x ∈ G andPx the law of the random walk started at x. If

∑y∼x c([x, y]) = 1, then

Ex[T+x ] = 2

∑e∈G

c(e).

Démonstration. Denote π the invariant measure associated with the conductances ofthe network. Then π(.)/π(G) is the invariant probability of the network and the meanreturn time formula yields

Eωx [T+

x ] =π(G)

π(x)= π(G),

since π(x) =∑

y∼x c([x, y]) = 1. Then we simply notice that

π(G) = 2∑e∈G

c(e).

Let us introduce another important random variable

S∞ = 2∞∑i=0

β−i(1 + Λi), (3.6)

which appears in the statement of our theorem. It is the mean return time to δ of thewalk on the innite tree T described in the algorithm following (3.3).

Lemma 3.2. There exists a constant Cψ depending on (pk)k≥0, such that for n ≥ 0 andk ≥ 0,

Q[ψn+1 = k] ≤ Cψkqk.

In particular, for another constant Cψ, supi∈N

EQ[ψi] ≤ Cψ <∞.

Démonstration. Recalling (3.3), we get

Q[ψn+1 = k] =k∑j=1

Q[φn+1 = j, ψn+1 = k]

= cnqk

k∑j=1

Q[Zn = 0]j−1Q[Zn+1 = 0]k−j

50


≤ cnkqk.

It is enough to show that the sequence (cn)n≥0 is bounded from above. A Galton-Watson tree of height n + 1 can be obtained as root having j children, one of whichproduces a Galton-Watson tree of height n, the others having no children of their own.Thu-shi2s

1/cn = Q[H = n+ 1]/Q[H = n] ≥ qjqj−10 ,

for any j ≥ 1. We x j0 ≥ 1 so that qj0 > 0 and we get

Q[ψn+1 = k] ≤ 1

qj0qj0−10

kqk−1,

where we used qk = pkqk−1 ≤ qk−1.

Using this lemma we can get a tail estimate for the height of traps.

Lemma 3.3. There exists α > 0 such that

Q[H ≥ n] ∼ αf ′(q)n.

Démonstration. It is classical (see [50]) that for any Galton-Watson tree of law Q withE eQ[Z1] = m < 1 expected number of children, we have

limn→∞

Q[Zn > 0]

mn> 0⇐⇒ E eQ[Z1 log+ Z1] <∞.

The integrability condition is satised for Q since qk = pkqk−1 ≤ qk−1, and the result

follows.

We also recall the following classical upper bound

Q[H ≥ n] = Q[Zn > 0] = Q[Zn ≥ 1] ≤ EQ[Zn] = f ′(q)n. (3.7)

The following lemma seems obvious, but not standard, so we include its proof forthe convenience of the reader.

Lemma 3.4. We have for k ≥ 0,

Q[Z1 ≤ k|Zn = 0] ≥ Q[Z1 ≤ k].

In particular EQ[Zi|Zn = 0] ≤ f ′(q)i, for any i ≥ 0 and n ≥ 0.

51


Démonstration. DenotingDn a geometric random variable of parameter 1−Q[Zn−1 = 0]which is independent of Z1, we have Q[Z1 ≤ k|Zn = 0] = Q[Z1 ≤ k|Z1 < Dn]. Thencompute

Q[Z1 ≤ k|Z1 < Dn] =

∑kj=0 Q[Dn > j]Q[Z1 = j]∑∞j=0 Q[Dn > j]Q[Z1 = j]

=

(1 +

∑∞j=k+1 Q[Dn > j]Q[Z1 = j]∑kj=0 Q[Dn > j]Q[Z1 = j]

)−1

,

now use that for all j′ < k < j we have Q[Dn > j] ≤ Q[Dn > k] ≤ Q[Dn > j′], yielding

Q[Z1 ≤ k|Z1 < Dn] ≥∑k

j=0 Q[Z1 = j]∑∞j=0 Q[Z1 = j]

= Q[Z1 ≤ k].

We can now estimate EQ[Λi].

Lemma 3.5. For all i ≥ 0,

EQ[Λi] ≤ Cψ1− (βf ′(q))−1

(f ′(q)β)i.

Démonstration. Using (3.5), Lemma 3.2 and Lemma 3.4, we get

EQ[Λi] = EQ[φi]EQ[Πi−1] + EQ[ψi − φi]EQ[Πi] ≤ EQ[Πi] supi∈N

EQ[ψi] ≤ Cψ

i∑j=1

f ′(q)jβj,

and the result follows immediately, since βf ′(q) > 1.

Finally, we get the following

Proposition 3.1. We have

EQ[S∞] ≤ 2Cψ1− (βf ′(q))−1

(β

β − 1+

1

1− f ′(q)

)<∞.

Démonstration. Recalling Lemma 3.5, we get

EQ[S∞] ≤ 2∞∑i=0

β−iEQ[1 + Λi] ≤ 2Cψ1− (βf ′(q))−1

∞∑i=0

β−i(1 + (βf ′(q))i) <∞,

and we conclude using f ′(q) < 1.

52

4. SKETCH OF THE PROOF

4 Sketch of the proof

In the rst step, we show (see Theorem 5.1) that the time is essentially spent inhn-traps.

Then we show that these hn-traps are far away from each other, and thu-shi2s thecorrelation between the time spent in dierent hn-traps can be neglected. Moreover thenumber of hn-traps met before level n is roughly ρCanε. Let

χ0(n) = the time spent in the rst hn − trap met (4.1)

where we point out that there can be several visits to this trap. At this point we havereduced our problem to estimating

∆n ≈ χ1(n) + · · ·+ χρCanε(n),

where χi(n) are i.i.d. copies of χ0(n).Now we decompose the time spent in the rst hn-trap according to the number of

excursions in it starting from the root

χ1(n) =Wn∑i=1

T(i)0 ,

whereWn denotes the number of visits of the trap until time n and T (i)0 an i.i.d. sequence

of random variables measuring the time spent during an excursion in a big trap. It isimportant to notice that the presence of an hn-trap at a vertex gives information onthe number of traps at this vertex, and thu-shi2s on the geometry of the backbone.So the law of Wn depends on n. Nevertheless we show that this dependence can beasymptotically neglected, and that for large n, Wn is close to some random variableW∞(Proposition 6.1).

Now we have essentially no more correlations between what happens on the backboneand on big traps. The only thing left to understand is the time spent during an excursionin a hn-trap from the root. To simplify if the walker does not reach the point δ in thetrap (this has probability ≈ 1− p∞), the time in the trap can be neglected. Otherwise,the time spent to go to δ, and to go directly from δ back to the root of the trap canalso be neglected, in other words, only the successive excursions from δ contribute tothe time spent in the trap. This is developed in Section 8, and we have

χ1(n) ≈Bin(W∞,p∞)∑

i=1

G(i)−1∑j=0

T (i,j)exc , (4.2)

where T (i,j)exc are i.i.d. random variables giving the lengths of the excursions from δ to δ.

Further, G(i) is the number of excursions from δ during the i-th excursion in the trap : it

53


is a geometric random variable with a parameter of order β−H . Since β−H is very small(H being conditioned to be big), the law of large numbers should imply that

G(i)−1∑j=0

T (i,j)exc ≈ G(i)Eω[T (i,j)

exc ] ≈ G(i)S∞,

and also we should have G(i) − 1 ≈ βHei. This explains why, recalling (1.3),

χ1(n) ≈ βHZ∞.We are then reduced to considering sums of i.i.d random variables of the form Ziβ

Xi

with Xi integer-valued. This is investigated in Section 10. We then nish the proof ofTheorem 1.3 in Section 11.

Remark 4.1. The reasoning fails in the critical case γ = 1, indeed in this case wehave to consider a critical height hn which is smaller. This causes many problems, inparticular in big traps there can be big subtraps and so, for example, the time to go fromthe top to the bottom of a trap cannot be neglected anymore.

5 The time is essentially spent in big traps

We recall that hn = d−(1− ε) lnn/ ln f ′(q)e. Lemma 3.3 gives the probability that atrap is an hn-trap :

ηn := Q[H ≥ hn] ∼ αf ′(q)hn , (5.1)

For x ∈ backbone, we denoteLx the set of traps rooted at x (5.2)

(if x is not in the backbone then Lx = ∅). Let us denote the vertices in big traps byL(hn) = y ∈ T (ω) | y is in a hn − trap..

Our aim in this section is to show the following

Proposition 5.1. For ε > 0, we have

for all t ≥ 0, P[∣∣∣∣∆n − χ(n)

n1/γ

∣∣∣∣ ≥ t

]→ 0

whereχ(n) = card1 ≤ i ≤ ∆n|Xi−1, Xi ∈ L(hn) (5.3)

is the time spent in big traps up to time ∆n.

Dene

54

5. THE TIME IS ESSENTIALLY SPENT IN BIG TRAPS

(i) A1(n) = ∆Yn ≤ C1n,

(ii) A2(n) = card∪∆Yn

i=1LYi ≤ C2n(iii) A3(n) =

max

`∈LYi ,i≤∆Yn

card0 ≤ i ≤ ∆Yn |Yi ∈ `, Xσi+1 ∈ ` ≤ C3 lnn

,

(iv) A(n) = A1(n) ∩ A2(n) ∩ A3(n).The following lemma tells us that typically the walk spends less than C1n time units

before reaching level n, sees less than C2n traps and enters each trap at most C3 lnntimes.

Lemma 5.1. For appropriate constants C1, C2 and C3, we have

P[A1(n)c] = o(n−2) and P[A(n)c]→ 0.

Démonstration. By a comparison to the β-biased random walk on Z, standard largedeviations estimates yields

P[A1(n)c] = o(n−2),

for C1 large enough.On A1(n), the number of dierent vertices visited by (Yi)i≥0 up to time ∆Y

n is atmost C1n. The descendants at each new vertex are drawn independently of the precedingvertices. Moreover at each vertex the mean number of traps at most the mean numberof children, thu-shi2s E[cardL0] ≤ m/(1 − q). The law of large numbers yields forC2 > C1m/(1− q) that

P[A2(n)c] ≤ P[C1n∑i=0

cardL(i)0 > C2n

]+ P[A1(n)c]→ 0,

where cardL(i)0 are i.i.d. random variables with the law of cardL0. This yields the second

part.For A3(n), we want, given a vertex x in the backbone and any ` ∈ Lx to give an

upper bound on the number of transitions from x to y, where y is the bud associated to`. Let z be an ospring of x in the backbone. Then, at each visit to x, either the walkerdoes not visit y or z, or it has probability 1/2 to visit y rst (or z rst). Hence,

(i) the number of transitions from x to y before reaching z is dominated by a geo-metric random variable of parameter 1/2,

(ii) the number of transitions from x to z is dominated by a geometric randomvariable of parameter p∞, since the escape probability from z is at least p∞.

Consequently the number of transitions from x to y is dominated by a geometricrandom variable of parameter p∞/2. Thu-shi2s

P[A3(n)c ∩ A2(n)] ≤ C2nP[G(p∞/2) ≥ C3 lnn

]≤ CnC3 ln(1−p∞/2)+1,

and if we take C3 large enough we get the result.

55


Now we can start proving Proposition 5.1. Decompose ∆n into

∆n = ∆Yn + χ(n) +

∑`∈ ∪∆Y

ni=0LYi \ L(hn)

N(`), (5.4)

where N(`) = 1 ≤ i ≤ ∆n|Xi−1 ∈ `,Xi ∈ `.The distribution of N(`) conditionally on the backbone, the buds and (Y ′i )i≤∆Y ′

n, the

walk on the backbone and the buds, is∑E`

i=1 R(i)` . Here we denoted E` the number of

visits to ` and R(i)` is the return time during the i-th excursion from the top of `. These

quantities are considered for traps `, conditioned to have height at most hn.Obviously we get from (5.4) that

∆n ≥ χ(n). (5.5)

From (5.4) we get for t > 0,

P[

∆n−χ(n)

n1/γ > t]≤ P[A(n)c] + P

[C1n+

∑C2ni=0

∑C3 lnnj=0 R

(i)` ≥ tn1/γ

]≤ o(1) + P

[∑C2ni=0

∑C3 lnnj=0 R

(i)` ≥ t

2n1/γ

],

(5.6)

where we used Lemma 5.1.Chebyshev's inequality yields,

P[C2n∑i=0

C3 lnn∑j=0

R(i)` ≥

t

2n1/γ

]≤ 2

tn1/γE[C2n∑i=0

C3 lnn∑j=0

R(i)`

]≤ 2C2C3n

1−1/γ lnn

tE[R

(1)1 ] .

Using Lemma 3.4 and Lemma 3.1, we have

E[R(1)1 ] = EQ[Eω

root[T+root]|H < hn] = 2

hn−1∑i=0

βiEQ[Zn|H < hn]

≤ 2hn−1∑i=0

(βf ′(q))i ≤ Cn(1−ε)(−1+1/γ).

Plugging this in the previous inequality, we get for any ε > 0 and t > 0

P[C2n∑i=0

C3 lnn∑j=0

R(i)` ≥

t

2n1/γ

]= o(1),

thu-shi2s recalling (5.6) and (5.5) we have proved Proposition 5.1.

56

6. NUMBER OF VISITS TO A BIG TRAP

6 Number of visits to a big trap

We denote Kx = max`∈Lx H(`), the height of the biggest trap rooted at x for x ∈backbone, where we recall that H denotes the height of the trap from the bud and notfrom the root.

Lemma 6.1. We haveP[K0 ≥ hn] ∼ Caf

′(q)hn ,

where Ca = αqm−f ′(q)1−q , recalling Lemma 3.3 for the denition of α.

Démonstration. We denote Z the number of children of a the root and Z∗ the numberof children with an innite line of descent. Let P be the law of a f -Galton Watson treewhich is not conditioned on non-extinction and E the corresponding expectation. Recall(5.1) and let H(i), i = 1, 2, . . . be i.i.d. random variables which have the law of the heightof a h-Galton-Watson tree. Then

P[K0 ≥ hn] = P[

maxi=1,...,Z−Z∗

H(i) ≥ hn

]= 1− E[(1− ηn)Z−Z

∗(1− 1Z∗ = 0)]

1− q ,

where the indicator function comes from the conditioning on non-extinction, whichcorresponds to Z∗ 6= 0.

Hence

P[K0 ≥ hn] = 1− E[(1− ηn)Z−Z∗]− E[(1− ηn)Z−Z

∗0Z∗]

1− q ,

and using E[sZ−Z∗tZ∗] = f(sq + t(1− q)) (see [65]) we get

P[K0 ≥ hn] = 1− f((1− ηn)q + 1− q)− f((1− ηn)q)

1− q .

Now, using (5.1) and the expansion f(z − x) = f(z) − f ′(z)x + o(x) for z ∈ q, 1,we get the result.

Dene the rst time when we meet the root of a hn-trap using the clock of Yn,

K(n) = infi ≥ 0|KYi ≥ hn. (6.1)

We also dene `(n) to be a hn-trap rooted at YK(n), if there are several possibilities wechoose one trap according to some predetermined order. We denote b(n) the associatedbud.

We describe, on the event 0 − SR, the number of visits to `(n), by the followingrandom variable :

Wn = cardi | Xi = YK(n), Xi+1 = b(n), (6.2)

where ω is chosen under the law P[·] and Xn under P ω0 [·|0 − SR]. We will need the

following bounds for the random variables (Wn)n≥1.

57


Lemma 6.2. We have Wn G(p∞/3) for n ∈ N, i.e. the random variables Wn arestochastically dominated by a geometric random variable with parameter p∞/3.

Démonstration. For n ∈ N, starting from any point x of the backbone, the walker hasprobability at least 1/3 to go to an ospring y of x on the backbone before going tob(n) or ←−x . But the rst hitting time of y has probability at least p∞ to be a super-regeneration time. The result follows as in the proof of Lemma 5.1.

Proposition 6.1. There exists a random variable W∞ such that

Wnd−→ W∞.

where we recall that for the law of Wn, ω is chosen under the law P[·] and Xn underP ω

0 [·|0− SR].

Remark 6.1. It follows from Lemma 6.2 that W∞ G(p∞/3).

Fix n ∈ N∗ and set m ≥ n. We aim at comparing the law of Wm with that of Wn

and to do that we want to study the behaviour of the random walk starting from thelast super-regeneration time before a hn-trap (resp. hm-trap) is seen. This motivates thedenition of the last super-regeneration time seen before time n,

Σ(n) := max0 ≤ i ≤ n | i− SR.For our purpose it is convenient to introduce a modied version of Wm, which will

coincide with high probability with it. For m ≥ n, recall that θ denotes the time-shiftfor the walk and set

K(m,n) = infj ≥ 0 | KYj ≥ hm, `(m) θΣ(j) = `(n) θΣ(j),the rst time the walker meets a hm-trap which is the rst hn-trap of the currentregeneration block and we denote by b(m,n) the associated bud. Set

Wm,n = cardi | Xi = YK(m,n), Xi+1 = b(m,n),where ω is chosen under the law P[·] and (Ui)i≤K(m,n) under P

ω0 [·|0-SR].

Lemma 6.3. For m ≥ n we have that

Wm,nd= Wn.

Démonstration. To reach a vertex where an hm-trap is rooted, the walker has to reacha vertex where an hn-trap is rooted. Two cases can occur : either the rst hn-trap metis also a hm-trap or it is not. In the former case, which has probability ηm/ηn > 0,since the height of the rst hn-trap met is independent of the sequence (Ui)i≤K(n), the

58


random variables Wm,n and Wn coincide. In fact they coincide with the number oftransitions from the backbone to the bud of the rst hn-trap met. In the latter case,by its denition, K(m,n) cannot occur before the next super-regeneration time, henceK(m,n) ≥ τ1 θK(n). In this case Wm,n = Wm,n θK(n) and then by Proposition 2.1,

Wm,n θτ1θK(n)

d= Wm,n,

and Wm,n θτ1θK(n)is independent of (Ui)i≤τ1θK(n)−1.

The scenario repeats itself until the hn-trap reached is in fact a hm-trap, the numberof attempts necessary to reach this hm-trap is a geometric random variable of parameterηm/ηn which is independent of the (Ui)'s.

This means that there is a family (W(i)n )i≥1 of i.i.d. random variables with the same

law as Wn such thatWm,n = W (G)

n ,

where G is a geometric random variable independent of the (W(i)n )i≥1. Then, note that

we haveWm,n = W (G)

nd= Wn.

Now we need to show that Wm,n and Wm coincide with high probability, so weintroduce the event

Am,n = `(m) = `(n) θΣ(K(m)),on which clearly Wm,n and Wm are equal.

Lemma 6.4.supm≥n

P[Acm,n|0− SR]→ 0 for n→∞. (6.3)

Démonstration. Let us denote, recalling (5.2)

V ij =

card

τ1⋃k=0

` ∈ LYk , ` is a hj-trap = i,

and

V i,+j =

card

τ1⋃k=0

` ∈ LYk , ` is a hj-trap ≥ i.

Then we haveP[Acm,n|0− SR] ≤ P[V 2,+

n |V 1,+m , 0− SR]. (6.4)

Let us denote cardTrap the number of traps seen before τ1,

cardTrap = card` | ` ∈

τ1⋃i=0

LYi

,

59


and its generating function by

ϕ(s) := E[scardTrap|0− SR

].

The probability of Am,n can be estimated with the following lemma, whose proof isdeferred.

Lemma 6.5. We have

∀m ≥ n, P[V 2,+n |V 1,+

m , 0− SR] ≤ ϕ′(1)− ϕ′(1− ηn).

Now we have E[cardTrap|0 − SR

]≤ E[τ1|0 − SR]E[cardL0] < ∞ because of Re-

mark 2.1 and hence ϕ′ is continuous at 1, and (6.3) follows from (6.4).

Applying Lemma 6.5, Lemma 6.3 and (6.3) we get,

P[Wm ≥ y|0− SR] = P[Am,n,Wm,n ≥ y|0− SR] + o(m,n)

= P[Wm,n ≥ y|0− SR] + o(m,n)

= P[Wn ≥ y|0− SR] + o(m,n),

where supm≥n o(m,n)→ 0 as n goes to innity.The law of a random variable W∞ can be dened as a limit of the laws of some

subsequence of (Wm), since the family (Wm)m≥0 is tight by Lemma 6.2. Then taking mto innity along this subsequence in the preceding equation yields

∀t > 0, P[W∞ ≤ t | 0− SR] = P[Wn ≤ t|0− SR] + o(1).

This proves Proposition 6.1.It remains to show Lemma 6.5.

Démonstration. Note that for i ≥ 1,

P[V in|V 1,+

m , 0− SR] =P[V i

n|0− SR]

P[V 1,+m |0− SR]

P[V 1,+m |V i

n, 0− SR] (6.5)

≤ P[V in|0− SR]

P[V 1,+m |0− SR]

iQ[H ≥ hm|H ≥ hn]

= iP[V i

n|0− SR]

ηn

ηm

P[V 1,+m |0− SR]

≤ iP[V i

n|0− SR]

ηn.

60


Then we have∑i≥2

iP[V in|0− SR]

=∑i≥2

∑j≥i

P[cardTrap = j|0− SR]i

(j

i

)ηin(1− ηn)j−i

=∑j≥0

jP[cardTrap = j|0− SR]

j∑i=2

(j − 1

i− 1

)ηin(1− ηn)j−i

=ηn∑j≥0

jP[cardTrap = j|0− SR]

j−1∑i=1

(j − 1

i

)ηin(1− ηn)(j−1)−i

=ηn∑j≥0

jP[cardTrap = j|0− SR](

1− (1− ηn)j−1)

=ηn(ϕ′(1)− ϕ′(1− ηn)).

Inserting this in (6.5) we get

P[V 2,+n |V 1,+

m , 0− SR] =∞∑i=2

P[V in|V 1,+

m , 0− SR]

≤ 1

ηn

∑i≥2

iP[V in|0− SR] = ϕ′(1)− ϕ′(1− ηn),

which concludes the proof of Lemma 6.5.

We will need the following lower bound for the random variable W∞.

Lemma 6.6. There exists a constant cW > 0 depending only on (pi)i≥0, such that

P[W∞ ≥ 1] ≥ cW .

Démonstration. By Proposition 6.1, it is enough to show the lower bound for all Wn.First let us notice that

P[Wn ≥ 1 | 0− SR] ≥ E

[(1

Z(K(n)) + 1

)2

p∞

]≥ (1− f ′(q)) E

[ 1

(Z(K(n)) + 1)2

],

(6.6)where Z(K(n)) is the number of ospring of YK(n). To show (6.6), note that the particlehas probability at least β/(βZ(K(n)) + 1) ≥ 1/(Z(K(n)) + 1) of going from YK(n) tob(n) and when it comes back to YK(n) again there is probability at least 1/(Z(K(n)) +1) to go from YK(n) to one of its descendants on the backbone and then there is a

61


probability of at least p∞ that a super-regeneration occurs. The event we just describedis in Wn ≥ 1 ∩ 0 − SR. For the second inequality in (6.6), use β > βc = f ′(q)−1

hence p∞ = 1− β−1 ≥ 1− f ′(q).Now, we notice that the law of the Z(K(n)) is that of Z1 conditioned on the event

an hn− trap is rooted at 0. Denote j0 the smallest index such that j0 > 1 and pj0 > 0(which exists since m > 1) and Z∗1 the number of descendants of 0 with an innite lineof descent. All Z1−Z∗1 traps rooted at 0 have, independently of each other, probabilityηn of being hn-traps, so that

P[Z(K(n)) = j0] = P[Z1 = j0 | an hn − trap is rooted at 0]

=P[Z1 = j0, Bin(Z1 − Z∗1 , ηn) ≥ 1]

P[an hn − trap is rooted at 0]

≥ ηnP[Z1 = j0, Z1 − Z∗1 ≥ 1]

P[an hn − trap is rooted at 0].

Further, since P [Bin(Z1 − Z∗1 , ηn) ≥ 1] ≤ Z1ηn, we have

P[an hn − trap is rooted at 0] ≤∞∑j=0

P[Z1 − Z∗1 = j]jηn ≤mηn.

Putting these equations together, we get that

P[Z(K(n)) = j0] ≥ P[Z1 = j0, Z1 − Z∗1 ≥ 1]

m.

The last equation and (6.6) yield a lower bound for P[W∞ ≥ 1] which depends onlyon (pk)k≥0.

7 The time spent in dierent traps is asymptotically

independent

In order to show the asymptotic independence of the time spent in dierent big trapswe shall use super-regeneration times. First we show that the probability that there isa hn-trap in the rst super-regeneration block goes to 0 for n→∞.

Dene(i) B1(n) = ∀i ∈ [1, n], cardYτi , . . . , Yτi+1

≤ nε,(ii) B2(n) = ∀i ∈ [0, τ1], cardLYi ≤ n2ε,(iii) B3(n) = ∀i ∈ [0, τ1], ∀` ∈ LYi , ` is not a hn-trap,(iv) B4(n) = ∀i ∈ [2, n], card

LYj | j ∈ [τi, τi+1], contains a hn-trap

≤ 1,(v) B(n) = B1(n) ∩B2(n) ∩B3(n) ∩B4(n).

62

7. THE TIME SPENT IN DIFFERENT TRAPS IS ASYMPTOTICALLYINDEPENDENT

Lemma 7.1. For ε < 1/4, we have

P[B1(n)c] = o(n−2) and P[B(n)c]→ 0.

Démonstration. Since τ2 − τ1 (resp. τ1) has some positive exponential moments andB1(n)c ⊆ ∪ni=1τi+1 − τi ≥ nε,

P[B1(n)c] = o(n−2).

Using the fact that the number of traps at dierent vertices has the same law,

P[B2(n)c] ≤ P[B1(n)c] + nεP[cardL0 ≥ n2ε] ≤ o(1) + n−εm

1− q = o(1),

where we used Chebyshev's inequality and E[cardL0] ≤ E[Z1] ≤m/(1− q).Then we have

P[B3(n)c] ≤ P[B2(n)c] + n3εηn = o(1),

yielding the result using (5.1), since ε < 1/4.Finally, up to time n we have at most n super-regeneration blocks, on B1(n) they

contain at most nε visited vertices. But the probability that among the nε rst vis-ited vertices after a super-regeneration time, two of them are adjacent to a big trap isbounded above by n2εP[K0 ≥ hn]2 (here we implicitely used Remark (2.1)). Hence, weget

P[B4(n)c] ≤ P[B1(n)c] + nn2ε(Cnε−1)2 = O(n4ε−1),

yielding the result for ε < 1/4.

We dene R(n) = cardY1, . . . , Y∆Yn and ln the number of vertices where an hn-trap

is rooted : ln = cardi ∈ [0,∆Y

n ], LYi contains a hn-trap. Recall (2.2) and dene

C1(n) = (1− n−1/4)ρn ≤ R(n) ≤ (1 + n−1/4)ρn (7.1)

C2(n) =

(1− n−ε/4)ρCanf′(q)hn ≤ ln ≤ (1 + n−ε/4)ρCanf

′(q)hn

(7.2)

C3(n) =∀1 ≤ i ≤ ∆Y

n , card` ∈ LYi |` is a hn-trap

≤ 1

(7.3)

and C(n) = C1(n) ∩ C2(n) ∩ C3(n).


P[C(n)c]→ 0.

63


Démonstration. First, we notice that for i ≥ 0 and with the convention τ0 := 0 we have

Zi := cardYτi+1, . . . , Yτi+1 G(i)(p∞),

where the geometric random variables G(i) are i.i.d. Indeed at each new vertex visitedwe have probability at least p∞ to have a super-regeneration time. Let us denote n0 thesmallest integer such that ∆Y

n ≤ τn0 , which satises n0 ≤ n since∣∣Xτi+1

∣∣ − |Xτi | ≥ 1.

Now, since the random variables Zi are i.i.d. andn0−1∑i=1

Zi ≤ cardY1, . . . , Y∆Yn ≤

n0∑i=1

Zi,

P[∣∣∣cardY1, . . . , Y∆Y

n

n− ρ∣∣∣ ≥ n−1/4

]≤n1/2Var

(cardY1, . . . , Y∆Yn

n

)=n1/2

(E[(cardY1, . . . , Y∆Y

n

n

)2]− E

[cardY1, . . . , Y∆Yn

n

]2)

≤n−3/2

(E[( n0∑

i=1

Zi

)2]− E

[n0−1∑i=1

Zi

]2)

≤n−1/2(E[G(p∞)2] + E[G(p∞)]2),

yielding P[C1(n)c]→ 0.On C1(n) we know that there are R(n) ∈ [ρn(1−n−1/4), ρn(1+n−1/4)] vertices where

we have independent trials to have hn-traps. Hence ln has the law Bin(R(n),P[K0 ≥hn]), where the success probability satises P[K0 ≥ hn] ≤ Cnε−1 has asymptoticsgiven by Lemma 6.1. Now, standard estimates for Binomial distributions imply thatP[C2(n)c ∩ C1(n)]→ 0.

On C2(n), there are at most Cnε vertices where (at least) one hn-trap can be rooted,we only need to prove that, with probability going to 1, those vertices do not containmore than two hn-traps. Using the same reasoning as in Lemma 6.5 we get

P[0 has at least two hn-traps|0 has at least one hn-trap, 0− SR]

≤ f ′(1)− f ′(1− ηn) ≤ Cηn,

where we used that E[Z2] <∞, which implies that f ′′(1) <∞.The result follows from the fact that ηn = o(n−ε) for ε < 1/4.

Let us denote, recalling (3.1),

D(n) =

max` ∈ ∪i=0,...,∆Y

nLYi

H(`) ≤ 2 lnn

− ln f ′(q)

.

64

8. THE TIME IS SPENT AT THE BOTTOM OF THE TRAPS

Lemma 7.3. We have

P[D(n)c]→ 0.

Démonstration. Due to (3.7), we know that Q[H ≥ 2 lnn− ln f ′(q)

] ≤ n−2, so using Lemma 5.1

P[D(n)c] ≤ P[A2(n)c] + P[A2(n) ∩D(n)c] ≤ o(1) + C2n−1 = o(1),

which concludes the proof.

On B(n) there is no big trap in the rst super-regeneration block, on B(n)∩C(n) allbig traps are met in distinct super-regeneration blocks and C2(n) tells us the asymptoticnumber of such blocks. Moreover on D(n), we know that to cross level n on a trap, ithas to be rooted after level n− (−2 lnn/ ln f ′(q)). Hence using Lemma 7.1, Lemma 7.2,Lemma 7.3, Proposition 2.1 and Remark 2.3, we get

Proposition 7.1. Let χi(n) be i.i.d. copies of χ0(n), see (4.1), andn = n− (−2 lnn/ ln f ′(q)). Then we have

(1−en−ε/4)ρnCaenf ′(q)hen∑i=1

χi(n) χ(n) (1+n−ε/4)ρnCanf ′(q)hn∑

i=1

χi(n).

In the light of Proposition 5.1, our problem reduces to understanding the convergencein law of a sum of i.i.d. random variables. The aim of the next section is to reduce χ1(n)to a specic type of random variable for which limit laws can be derived (see Section 10).

8 The time is spent at the bottom of the traps

We denote by δi(n) (resp. rooti(n), bi(n)) the leftmost bottom point (the root, thebud) of the i-th hn-trap seen which is called `j(n). In a similar fashion χi denotes thetime spent in the i-th hn-trap met.

We want to show that the time spent in the big traps is essentially spent at thebottom of them, i.e. during excursions from the the bottom leftmost point δ. In orderto prove our claim, we introduce

χ∗j(n) = cardk ≥ 0 | Xk ∈ `j(n), k ≥ Tδj(n), Tδj(n) θk <∞,

the time spent during excursions from the bottom in the j-th hn-trap met. It is obviousthat

χj(n) ≥ χ∗j(n).

We prove that

65


Proposition 8.1. For ε < 1/4, we have that

for all t > 0, P

1

n1/γ

∣∣∣ l(n)∑j=1

(χj(n)− χ∗j(n)

)∣∣∣ ≥ t

→ 0.

In order to prove the preceding theorem, we mainly need to understand χ1(n) andχ∗1(n). Note that χ1(n) is a sum of Wn successive i.i.d. times spent in `(n) and χ∗1(n)is a sum of Wn successive i.i.d. times spent during excursions from the bottom of `(n).We can rewrite the theorem as follows

for all t > 0, P[ 1

n1/γ

∣∣∣ l(n)∑i=1

W(i)n∑

j=1

(T jrooti(n) − T ∗,jrooti(n))∣∣∣ ≥ t

]→ 0 (8.1)

where

T jrooti(n) = k ≥ 0 | Xk ∈ ì(n), cardk ≤ k, Xek+1 = bi(n), Xek = rooti(n) = j,

and

T ∗,jrooti(n) = k ≥ 0 | Xk ∈ ì(n), cardk ≤ k, Xek+1 = bi(n), Xek = rooti(n) = j,

k ≥ Tδj(n), Tδj(n) θk <∞,

and (W(i)n ), i ≥ 1 are i.i.d. copies of Wn.

Consequently, in this section we mainly investigate the walk on a big trap, whichis a random walk in a nite random environment. Recall that root is the vertex YK(n)

on the backbone where `(n) is attached. Moreover set Qn[·] = Q[· | H ≥ hn], EQn [·] =EQ[· | H ≥ hn], Eω[·] := Eω

root[·] and EQn [·] = EQn [Eω[·]].Remark 8.1. To ease notations, we add to all these probability spaces an independentrandom variable Wn whose law is given by (6.2), under the law P[·|0 − SR] for n ∈N ∪ ∞.

We will extensively use the description of Section 3, in particular we recall that atrap is composed of root which is linked by an edge to a h-Galton-Watson tree.

We want to specify what `(n) looks like. Denoting

h+n =

⌈(1 + ε) lnn

− ln f ′(q)

⌉,

consider(i) A1(n) = H ≤ h+

n ,(ii) A2(n) = there are fewer than nε subtraps,

66


(iii) A3(n) = all subtraps of `(n) have height ≤ hn,(iv) A(n) = A1(n) ∩ A2(n) ∩ A3(n).


Qn[A(n)c] = o(n−ε).

Démonstration. First

Qn[A1(n)c] ≤ Q[H ≥ h+n ]

Q[H ≥ hn]≤ Cn−2ε = o(n−ε).

Furthermore using Lemma 3.2, Lemma 3.4 and (3.7), we get

Qn[A2(n)c]

≤Qn[A1(n)c] + Qn[A1(n), there are nε/h+n subtraps on a vertex of the spine]

≤o(n−ε) + h+nCψ

nε

h+n

qnε/h+

n = o(n−ε).

Finally

Qn[A3(n)c]

≤Qn[A2(n)c] + Qn[A2(n), there exists a subtrap of height ≥ hn]

≤o(n−ε) + nεηn = o(n−ε),

where we used (5.1) and ε < 1/4.

Using Chebyshev's inequality we get, recalling (7.2),

P[ 1

n1/γ

∣∣∣ l(n)∑i=1

W(i)n∑

j=1

(T jrooti(n) − T ∗,jrooti(n))∣∣∣ ≥ t

]

≤ 1

tn1/γE[1C2(n)1A(n)

l(n)∑i=1

W(i)n∑

j=1

(T jrooti(n) − T ∗,jrooti(n))]

+ P[C2(n)c] + Qn[A(n)c]

≤ P[C2(n)c] + Qn[A(n)c] +2ρCacβn

ε

tn1/γE[1A(n)(T 1

root1(n) − T ∗,1root1(n))],

where cβ = E[G(p∞/3)], implying E[W(j)n ] ≤ cβ. Hence using Lemma 8.1 and

Lemma 7.2

P[ 1

n1/γ

∣∣∣ l(n)∑i=1

W(i)n∑

j=1

(T jrooti(n)−T ∗,jrooti(n))∣∣∣ ≥ t

]−Ctnε−1/γEQn [1A(n)(T 1

root1(n)−T ∗,1root1(n))]→ 0

(8.2)

67


We have to estimate this last expectation. Consider an hn-trap. Each time the walkerenters the hn-trap two cases can occur : either the walker will reach δ, or he will notreach δ before he comes back to root. In the former case, T+

root − T ∗,+root is the time spentgoing from root to δ for the rst time plus the time coming back from δ to root for thelast time (starting from δ and going back to root without returning to δ). In the lattercase, T+

root − T ∗,+root equals T+root. This yields the following upper bound

E[1A(n)(T+root − T ∗,+root)] (8.3)

≤EQn [1A(n)Eωroot[1A(n)T+

δ | T+δ < T+

root]] + EQn [1A(n)Eωδ [T+

root | T+root < T+

δ ]]

+ EQn [1A(n)Eωroot[T

+root | T+

root < T+δ ]].

To tackle the conditionings that appear, we shall use h−processes, see [35] and [104]for further references. For a given environment ω let us denote hω the voltage (see [70])on the trap, given by hω(z) = P ω

z [T+root < T+

δ ], with hω(root) = 1 and hω(δ) = 0. Thenwe have the following formula for the transition probabilities of the conditioned Markovchain

P ωy [X1 = z|T+

root < Tδ] =hω(z)

hω(y)P ωy [X1 = z], (8.4)

for y, z in the trap.We recall that the voltage is harmonic except on δ and root. It can be computed

using electrical networks :

hω(y) = hω(y ∧ δ) = β−d(root,y∧δ) 1− β−(H+1−d(root,y∧δ))

1− β−(H+1).

In particular, comparing the walk conditioned on the event T+δ > T+

root to theoriginal walk, we have the following :

1. the walk remains unchanged on the subtraps,

2. for y on the spine and z a descendant of y not on the spine, we have

P ωy [X1 =←−y |T+

root < Tδ] > P ωy [X1 = z|T+

root < Tδ],

3. for y /∈ δ, root on the spine, we have

P ωy [X1 =←−y |T+

root < Tδ] > βP ωy [X1 = −→y |T+

root < Tδ].

The points (2) and (3) state respectively that the conditioned walk is more likelyto go towards root than to go to a given vertex of a subtrap and that restricted to thespine the conditioned walk is more than β-drifted towards root.

Lemma 8.2. For z ∈ δ, root, we have

EQn [1A(n)Eωz [T+

root | T+root < T+

δ ]] ≤ C(lnn)n(1−ε)(1/γ−1)+ε.

68


Démonstration. First let us show that the walk cannot visit too often a vertex of thespine. Indeed let y be a vertex of the spine, using fact (3), we have P ω

y [T+y > T+

root|T+root <

Tδ] ≥ p∞. Hence the random variable N(y) = cardn ≤ T+root| Xn = y with (Xn)

conditioned on T+root < Tδ is stochastically dominated by G(p∞), a geometric random

variable with parameter p∞.Furthermore, we cannot visit often a given subtrap s(y) ∈ Sy (recall 3.4). Indeed, if

we denote the number of visits to s(y) by N(s(y)) = cardn ≤ T+root| Xn = y, Xn+1 ∈

s(y), using remark (2) and a reasoning similar to the one for the asymptotics on A3(n) inLemma 5.1 we have that N(s(y)) with (Xn) conditioned on T+

root < Tδ is stochasticallydominated by G(p∞/2).

Let us now consider the following decomposition

T+root = Tspine +

∑s∈subtraps

N(s)∑j=1

Rjs,

where Tspine = cardn ≤ T+root|Xn is in the spine =

∑x∈spineN(x), and Rj

s is the timespent in the subtrap s during the j-th excursion in it. Moreover, on A(n), the law ofany subtrap s is that of a Galton-Watson tree conditioned to have height strictly lessthan i + 1 for some i ≤ hn. Then Lemma 3.1 implies that for such a subtrap, Eω[Rj

s]has the same law as 2Πi,1,1

i+1 which satises using Lemma 3.4 that EQn [Πij] ≤ C(βf ′(q))i.

Moreover, on A(n), there are at most h+n vertices in the spine and at most nε subtraps,

hence

EQn [1A(n)Eωδ [T+

root | T+root < T+

δ ]] ≤ h+nE[G(p∞)]

+ h+nn

εE[G(p∞/2)]C(βf ′(q))hn ,

and using (βf ′(q))hn ≤ Cn(1−ε)(1/γ−1) we get

EQn [1A(n)Eωδ [T+

root | T+root < T+

δ ]] ≤ C(lnn)n(1−ε)(1/γ−1)+ε.

The previous proof is mainly based on the three statements preceding the statementof Lemma 8.2. Similarly, one can show the following

Lemma 8.3. For z ∈ δ, root, we have

EQn [1A(n)Eωz [T+

δ | T+δ < T+

root]] ≤ C(lnn)n(1−ε)(1/γ−1)+ε.

Démonstration. To apply the same methods as in the proof of Lemma 8.2, we only needthat the h-process corresponding to the conditioning on the event T+

δ < T+root satises

that

69


1. the walk remains unchanged on the subtraps,

2. for y on the spine and z a descendant of y not on the spine, we have P ωy [X1 =−→y |T+

δ < T+root] > P ω

y [X1 = z|T+δ < T+

root],

3. for y 6= δ, root on the spine, we have P ωy [X1 = −→y |T+

δ < T+root] > βP ω

y [X1 =←−y |T+δ < T+

root].

This immediately follows from the computation of the voltage hω, given by hω(z) =

P ωz [T+

δ < T+root], with h

ω(root) = 0 and hω(δ) = 1. A computation gives

hω(y) = hω(d(y ∧ δ, δ)) =βH+1 − βd(y∧δ,δ)

βH+1 − 1. (8.5)

From (8.3), Lemma 8.3 and Lemma 8.2, we deduce that

E[1A(n)(T+root − T ∗,+root)] ≤ C(lnn)n(1−ε)(1/γ−1)+ε (8.6)

Now using (8.6) and (8.2) we prove (8.1), more precisely

for all t > 0, P

[∣∣∣∣∣∑l(n)

i=1 χj(n)− χ∗j(n)

n1/γ

∣∣∣∣∣ ≥ t

]≤ o(1) + C(lnn)n2ε−1−ε(1/γ−1),

and thu-shi2s Proposition 8.1 follows for ε < 1/4.

9 Analysis of the time spent in big traps

Let us denote Qn := Q[ · | H = h0n] where

h0n = dlnn/− ln f ′(q)e .

Note that

p1(H) := P ωδ [T+

δ < T+root] =

1− β−1

1− β−(H+1)(9.1)

where we recall that the distance between root and δ is 1 +H. Moreover let us denote

p2(H) := P ωδ [T+

root < T+δ ] =

1− β−1

βH − β−1. (9.2)

We have the following decomposition

χ∗1(n) =

Bin(Wn,p1(H))∑i=1

G(p2(H))(i)−1∑j=1

T (i,j)exc , (9.3)

70

9. ANALYSIS OF THE TIME SPENT IN BIG TRAPS

where T (i,j)exc is the time spent during the j-th excursion in the i-th trap, which is dis-

tributed under Qn as T+δ under P ω

δ [ · | T+δ < T+

root] with ω chosen according to Qn, forall (i, j). The T (i,j)

exc are independent with respect to P ω and for i1 6= i2 (T(i1,j)exc )j≥1 and

(T(i2,j)exc )j≥1 are independent with respect to Qn. For k ∈ Z and n large enough, let Zkn be a

random variable with the law of χ∗1(n)/βH under Qn+k and Zkn be a random variable withthe law of χ∗1(n)/βH under Qn+k. Furthermore we dene Z∞ := S∞

1−β−1

∑Bin(W∞,p∞)i=1 ei,

see (1.3), where (ei)i≥1 is a family of i.i.d. exponential random variables of parameter1, chosen independently of the (independent) random variables S∞ and W∞. Our aimis to show the following

Proposition 9.1. We have

Zkn d−→ Z∞.Moreover there exists a random variable Zsup such that

E[Z1−εsup ] <∞, for any ε > 0,

andfor n ∈ N and k > −n, Zkn Zsup.

Let us start by proving the convergence in law. The decomposition (9.3) for χ∗1(n)can be rewritten using (9.2)

χ∗1(n) = βHBin(Wn,p1(H))∑

i=1

1− β−H−1

1− β−1

βH − 1

βH − β−1

p2(H)

1− p2(H)

G(p2(H))(i)−1∑j=1

T (i,j)exc , (9.4)

which yields an explicit expression of Zkn. We point out that E[G(p2(H)) − 1] = (1 −p2(H))/p2(H). The convergence in law is due to the following facts (more precise state-ments follow below) :

1. For H large,(1− β−H−1)(βH − 1)

(1− β−1)(βH − β−1)≈ 1

1− β−1,

2. By the law of large numbers, we can expect

G(p2(H))(i)−1∑j=1

T (i,j)exc ≈ (G(p2(H))(i) − 1)Eω

δ [Texc],

3. Since p2(H) is small, (G(p2(H))(i) − 1)/E[G(p2(H))(i) − 1] ≈ ei,

4. Eωδ [T

(1,1)exc ] ≈ S∞ for H large enough,

71


5. Bin(Wn, p1(H)) ≈ Bin(W∞, p∞) sinceWnd−→ W∞ by Proposition 6.1 and p1(H)→

p∞ as H goes to innity.

Fact (1) is easily obtained, since for ξ > 0

Qn

[(1− ξ) 1

1− β−1≤ 1− β−H+1

1− β−1

βH − 1

βH − β−1≤ 1

1− β−1

]= 1, (9.5)

for n large enough.We start by computations with the measure Qn and we will be able to come back toQn+k.

For (2) and (4), we need to understand P ωδ [·|T+

δ < T+root] and to this end we will

consider the h-process associated with this conditioning. Recall the voltage hω given ashω(z) = P ω

z [Tδ < Troot], with hω(δ) = 1 and hω(root) = 0, see (8.5).We shall enumerate the vertices of the backbone from 0 to H + 1, starting from δ

up to root. With these new notations formula (8.5) becomes

hω(y) = hω(y ∧ δ) =βH+1 − βy∧δβH+1 − 1

, (9.6)

where y∧δ is identied to its number which is d(y∧δ, δ) as it is a vertex of the backbone.The transition probabilities are then given as in (8.4). Obviously they arise from

conductances, we may take(i) c(0, 1) = 1,

(ii) c(i, i+ 1) = c(i− 1, i)Pωi [X1=i+1|T+

δ <T+root]

Pωi [X1=i−1|T+δ <T

+root]

, for 1 ≤ i ≤ H,

(iii) c(i, z) = c(i, i − 1)Pωi [X1=z|T+

δ <T+root]

Pωi [X1=i−1|T+δ <T

+root]

, for i 6= 0 on the spine and z one of its

descendants which is not on the spine,(iv) c(y, z) = βc(←−y , y) for any vertex y not on the spine and z one of its descendants.We can easily deduce from this that for y 6= root in the trap and denoting z0 =

δ, . . . , zn = y the geodesic path from δ to y :

c(zn−1, y) =n−1∏j=1

P ωzj

[X1 = zj+1|T+δ < T+

root]

P ωzj

[X1 = zj−1|T+δ < T+

root],

which gives using (9.6) that

c(i, i+ 1) = β−ih(i+ 1)h(i)

h(1)h(0)= β−i

(1− βi−H)(1− βi−(H+1))

(1− β−H)(1− β−(H+1)). (9.7)

For a vertex z not on the spine, we have

c(z,←−z )= βd(←−z ,z∧δ) h(z ∧ δ)h(z ∧ δ − 1)

c(z ∧ δ, z ∧ δ − 1)

72


= βd(←−z ,z∧δ) 1− βz∧δ−(H+1)

1− βz∧δ−1−(H+1)c(z ∧ δ, z ∧ δ − 1). (9.8)

Together with Lemma 3.1, this yields, with Texc a generic random variable with thelaw of T (1,1)

exc ,

Eωδ [Texc] = 2

H−1∑i=0

β−i(1− βi−H)(1− βi−(H+1))

(1− β−H)(1− β−(H+1))

(1 +

1− βi−(H+1)

1− β(i−1)−(H+1)Λi(ω)

)(9.9)

where Λi was dened in (3.5).We see that the random variable S∞ is the limit of the last quantity as H goes to

innity. More precisely, using (9.9) we have 0 ≤ S∞ − Eωδ [Texc] and for n large enough

such that

for all k ≤ hn/2,(1− βk−hn)(1− βk−(hn+1))

(1− β−hn)(1− β−(hn+1))≥ 1− 2βhn ,

we get

S∞ − Eωδ [Texc] ≤ 2

(hn/2∑i=0

β−i(

1− (1− βi−H)(1− βi−(H+1))

(1− β−H)(1− β−(H+1))

)(1 + Λi)

)+ 2( ∞∑i=hn/2+1

β−i(1 + Λi)),

≤ 4β−hn/2(hn/2∑i=0

β−i(1 + Λi))

+ 2( ∞∑i=hn/2+1

β−i(1 + Λi)).

Hence, since S∞ ≥ 1 and using Chebyshev's inequality, we get

Qn

[(1− ξ)S∞ < Eω

δ [Texc] < S∞

]≥ 1−Qn[S∞ − Eω

δ [Texc] ≥ ξ] (9.10)

≥ 1− 1

ξβ−hn/2

10

1− β−1supi≥0

EQ[1 + Λi]

= 1 + o(1),

where we used Lemma 3.5 and the fact that ε < 1/4, this proves (4).In order to prove (2), we have to bound Eω

δ [T 21 ] from above. This is not possible for

all ω, but we consider the event

A5(n) =Eωδ [T 2

1 ] ≤ n1−2εγ

,

and show that it satises the following.

73


Lemma 9.1. For 0 < ε < min(1/3, 2γ/3), we have

Qn[A5(n)c]→ 0 .

Démonstration. In this proof we denote for y in the trap, N(y) the number of visits toy during an excursion from δ, which is distributed as card0 ≤ n ≤ T+

δ |Xn = y underP ωδ [·|T+

δ < T+root]. We have

Eωδ [T 2

exc] = Eωδ

[( ∑y∈trap

N(y))2]

≤∑

y,z∈trap

Eωδ [N(y)2]1/2Eω

δ [N(z)2]1/2

=( ∑y∈trap

Eωδ [N(y)2]1/2

)2

.

Now x y in the trap, denote q1 = P ωδ [T+

y < T+δ |T+

δ < T+root] and q2 = P ω

y [T+δ <

T+y |T+

δ < T+root]. Then we have

∀k ≥ 1, P ωδ [N(y) = k] = q1(1− q2)k−1q2.

Hence

Eωδ [N(y)2] =

∑n≥1

n2q1(1− q2)n−1q2 = q12− q2

q22

≤ 2q1

q22

.

Then by reversibility of the walk, if π is the invariant measure associated with theconductances c, we get q1 = π(δ)q1 = π(y)q2. This yields

Eωδ [N(y)2] ≤ 2π(y)

q2

. (9.11)

Furthermore we have

q2 ≥ (1/(Z1(y)β + 1))p∞β−d(δ,δ∧y)/2. (9.12)

Indeed suppose that y is not on the spine, otherwise the bound is simple. Startingfrom y, we reach the ancestor of y with probability at least (1/(Z1(y)β + 1)) then thewalker has probability at least β−d(y,y∧δ) to reach y ∧ δ before y, next he has probabilityat least 1/2 to go to

−−→y ∧ δ before going to z, where z is the rst vertex on the geodesic

path from y ∧ δ to y. Finally from−−→y ∧ δ, the walker has probability at least p∞ to go to

δ before coming back to−−→y ∧ δ.

We denote by π the invariant measure associated with the β-biased random walk(i.e. not conditioned on T+

δ < T+root), normalized so as to have π(δ) = 1. Then we have

74


1. For any y in the trap, π(y) ≤ π(y) because of (9.7) and (9.8),

2. and by denition of the invariant measure (Z1(y)β + 1)βd(δ,δ∧y)−d(y,δ∧y) = π(y).

Now plugging (2) in (9.12) yields a lower bound on q2 which can be used togetherwith (1) in (9.11) to get

Eωδ [N(y)2] ≤ Cβd(δ,δ∧y)π(y)2,

andEωδ [T 2

exc]1/2 ≤ C

∑y∈trap

βd(δ,δ∧y)/2π(y).

As a consequence, with A(n) as in Lemma 8.1 we get

EQn [1A(n)Eωδ [T 2

exc]12 ] ≤ CEQn

[1A(n)

h+n∑

i=0

β−i/2Λi

]

≤ C

h+n∑

i=0

(β1/2f ′(q))i

≤ C max(1, (β1/2f ′(q))h+n ),

where we used Lemma 3.1 and Lemma 3.4 for the rst inequality.Since (β1/2f ′(q))h

+n = n(1+ε)(1/2γ−1), we get by Chebyshev's inequality that

Qn[1A(n)Eωδ [T 2

exc]1/2 ≥ n

1−2ε2γ ] ≤ 1

n1−2ε2γ

EQn [1A(n)Eωδ [T 2

1 ]1/2]

≤ C max(n−1−2ε2γ , n3ε/(2γ)−1−ε).

The conditions on ε ensures that this last term goes to 0 for n→∞. Hence

P[A(n) ∩ A5(n)c]→ 0

and the result follows using Lemma 8.1.

We now turn to the study of

p2(H)

1− p2(H)

G(p2(H))−1∑i=1

T (i)exc.

Consider the random variable

Ng =⌊ −1

ln(1− p2(H))e⌋, (9.13)

where e is an exponential random variable of parameter 1. A simple computation showsthat Ng has the law of G(p2(H))− 1.

75


Set ξ > 0, we have using Chebyshev's inequality,

Qn

[(1− ξ)NgE

ωδ [Texc] ≤

Ng∑i=1

T (i)exc ≤ (1 + ξ)NgE

ωδ [Texc]

]≥1−Qn

[∣∣∣∑Ngi=1 T

(i)exc

Ng

− Eωδ [Texc]

∣∣∣ > ξEωδ [Texc], Ng 6= 0, Eω

δ [T 2exc] ≤ n(1−2ε)/γ

]−Qn[Eω[T 2

exc] ≥ n(1−2ε)/γ]−Qn[Ng = 0]

≥EQn

[n(1−2ε)/γ

Ng

1Ng 6= 0 1

ξ2

]−Qn[Eω[T 2

exc] ≥ n(1−2ε)/γ]−Qn[Ng = 0].

We have Qn[Ng = 0] = p2(H) ≤ p2(hn) ≤ Cn−(1−ε)/γ lnn, and hence

EQn

[1Ng 6= 0Ng

]= E

[− p2(H)

1− p2(H)ln p2(H)

]≤ Cn−(1−ε)/γ(lnn)2.

Putting together the two previous equations, using Lemma 9.1, we get for ξ < 1,

Qn

[(1− ξ)NgE

ωδ [Texc] <

Ng∑i=1

T (i)exc < (1 + ξ)NgE

ωδ [Texc]

]→ 1 . (9.14)

This shows (2). Turning to prove (3), we have

Qn

[(1− ξ)

⌊ 1

− ln(1− p2(H))e⌋≤ 1− p2(H)

p2(H)e ≤ (1 + ξ)

⌊ 1

− ln(1− p2(H))e⌋]

≥1−Qn

[(1− p2(H)

p2(H)− 1− ξ− ln(1− p2(H))

)e < 1

]−Qn

[(1− p2(H)

p2(H)− 1 + ξ

− ln(1− p2(H))

)e > −2

],

furthermore since∣∣∣1−pp − 1

− ln(1−p)

∣∣∣ is bounded on (0, ε1) by a certain M > 0 so that for

n large enough with p2(hn) < ε1, we get

Qn

[(1− ξ)

⌊ 1

− ln(1− p2(H))e⌋≤ 1− p2(H)

p2(H)e ≤ (1 + ξ)

⌊ 1

− ln(1− p2(H))e⌋]

(9.15)

≥1−Qn

[( ξ

− ln(1− p2(H))−M

)e < 1

]−Qn

[(− ξ

− ln(1− p2(H))+M

)e > −2

]≥ exp

(− 2

ξ/(− ln(1− p2(hn)))−M)≥ 1− (C/ξ)p2(hn),

which shows (3).

76


As a consequence of (9.10), (9.14) and (9.15), we see that for all ξ ∈ (0, 1),

Qn

[(1− ξ)S∞e ≤ p2(H)

1− p2(H)

Ng∑i=1

Eω[T (i)exc] ≤ (1 + ξ)S∞e

]→ 1 (9.16)

for n→∞. Using (9.2), (9.16) and (9.5) we get

Qn

[(1− ξ) S∞e

1− β−1≤ 1

βH

Ng∑i=1

Eω[T (i)exc] ≤ (1 + ξ)

S∞e

1− β−1

]→ 1 (9.17)

for n → ∞, which sums up (2), (3) and (4). For any k > −n, the equation (9.17)obviously holds replacing n with n+ k, and sinceQn[H = dln(n+ k)/ ln f ′(q)e] ≥ ck > 0 (this follows from Lemma 3.3), we have

Qn+k

[(1− ξ) S∞e

1− β−1≤ 1

βH

Ng∑i=1

Eω[T (i)exc] ≤ (1 + ξ)

S∞e

1− β−1

]→ 1. (9.18)

Only part (5) remains to be shown. Coupling Bin(Wn, p∞) and Bin(Wn, p1(H)) inthe standard way,

Qn+k[Bin(Wn, p∞) 6= Bin(Wn, p1(H))]

≤∑j≥0

P[Wn = j]Qn+k[Bin(j, p∞) 6= Bin(j, p1(H))]

≤∑j≥0

P[Wn = j]j(p1(h0

n+k)− p∞))

≤E[Wn](p1(h0

n+k)− p∞)

≤C (p1(h0n+k)− p∞

)→ 0, for n→∞

where C := E[G(p∞/3)] ≥ E[Wn] by Lemma 6.2. Hence,

Qn+k

[ 1

βH

Bin(Wn,p1(H))∑i=1

G(p2(H))−1∑j=1

T (i,j)exc ≥ t

]−Qn+k

[ 1

βH

Bin(Wn,p∞)∑i=1

G(p2(H))−1∑j=1

T (i,j)exc ≥ t

]→ 0

For ε1 > 0, introduceN(ε1) such that maxn≤∞ P[Wn ≥ N(ε1)] ≤ (1−p∞/3)N(ε1) ≤ ε1

and using the independence of Wn (for n ∈ N ∪ ∞) of the trap and the walk on thetrap, we get for any ε1 > 0,∣∣∣∣∣∣Qn+k

[ 1

βH

Bin(Wn,p∞)∑i=1

G(p2(H))−1∑j=1

T (i,j)exc ≥ t

]−Qn+k

[ 1

βH


G(p2(H))−1∑j=1

T (i,j)exc ≥ t

]∣∣∣∣∣∣77


≤∣∣∣∣∣∣∑j≥0

(P[Wn = j]− P[W∞ = j])Qn+k

[ 1

βH

Bin(j,p∞)∑i=1

G(p2(H))−1∑j=1

T (i,j)exc ≥ t

]∣∣∣∣∣∣≤∣∣∣∣∣∣∑

j∈[0,N(ε1)]

(P[Wn = j]− P[W∞ = j])Qn+k

[ 1

βH

Bin(j,p∞)∑i=1

G(p2(H))−1∑j=1

T (i,j)exc ≥ t

]∣∣∣∣∣∣+ ε1

and the right-hand side goes to ε1 as n goes to innity since

maxj≤N(ε1)

|P[Wn = j]− P[W∞ = j]| → 0,

by Proposition 6.1. So letting ε1 go to 0, we see that

Qn+k

[ 1

βH

Bin(Wn,p1(H))∑i=1

G(p2(H))−1∑j=1

T (i,j)exc ≥ t

]−Qn+k

[ 1

βH


G(p2(H))−1∑j=1

T (i,j)exc ≥ t

]→ 0

(9.19)Let us introduce

A(ξ) =for all i ∈ [1,Bin(W∞, p∞)],

1

βH

G(i)(p2(H))−1∑j=1

T (i,j)exc ∈

[(1− ξ) S∞

1− β−1ei, (1 + ξ)

S∞1− β−1

ei

],

where (ei)i≥1 is a sequence of i.i.d. exponential random variables of parameter 1 whichsatisfy

G(i)(p2(H))− 1 =⌊ −1

ln(1− p2(H))ei

⌋.

We have, denoting o1(1) the left hand side of (9.19)

Qn+k[A(ξ)] ≥∑i≥0

P[Bin(W∞, p∞) = i](1− o1(1))i

≥∑i≥0

P[Bin(W∞, p∞) = i](1− io1(1))

= 1− E[W∞]o1(1)→ 1 for n→∞ .

Hence, for any ξ > 0, we get

Qn+k

[ 1

βH

Bin(Wn,p1(H))∑i=1

G(p2(H))−1∑j=1

T (i,j)exc ≥ t

]− P

[ S∞1− β−1


ei ≥ t

1 + ξ

]→ 0

78


and

Qn+k

[ 1

βH

Bin(Wn,p1(H))∑i=1

G(p2(H))−1∑j=1

T (i,j)exc ≥ x

]− P

[ S∞1− β−1


ei ≥ t

1− ξ]→ 0

Concluding by using the two previous equations with ξ going to 0, we have thefollowing convergence in law :

Zkn =1

βH

Bin(Wn,p1(H))∑i=1

G(p2(H))−1∑j=1

T (i,j)exc

d−→ S∞1− β−1


ei,

where we recall that Zkn has the law of χ∗1(n)/βH under Qn+k and the ei are i.i.d. expo-nential random variables of parameter 1. This shows the rst part of Proposition 9.1.

Now let us prove the stochastic domination part. First notice that

Bin(Wn, p1(H)) G(p∞/3) and Eωδ [T1] T∞exc,

where T∞exc is distributed as the return time to δ, starting from δ, on an innite trap.Hence for k > −n

Zkn 1

βh0n+k

G(p∞/3)∑i=1

G(p2(h0n+k))∑

j=1

T∞,(i,j)exc ,

where (T∞,(i,j)exc )i,j≥1 are i.i.d. copies of T∞exc. Now recalling that

∑G(a)i=1 G(b)(i) has the

same law as G(ab), where all geometric random variables are independent, and usingthe fact that

βh0n+k ≥ cE[G(p∞p2(h0

n+k)/3)],

for some c = c(β) > 0, we get

Zkn C

E[G(p∞p2(h0n+k)/3)]

G(p∞p2(h0n+k)/3)∑

i=1

T∞,(i)exc .

Now, we prove the following technical lemma

Lemma 9.2. Let (Xi)i≥0 a sequence of i.i.d. non-negative random variables such thatE[X1] < ∞ and set Yi := (X1 + · · · + Xi)/i. Then there exists a random variable Ysup

such that

for all i ≥ 0, Yi Ysup,

and E[Y 1−εsup ] <∞ for all ε > 0.

79


Démonstration. Using Chebyshev's inequality we get that for any i ≥ 0,

for all t ≥ 0, P [Yi > t] ≤ 1

tE[X1].

If we choose Ysup such that P [Ysup > t] = min(1, E[X1]/t) for x ≥ 0, then Ysup stochas-tically dominates all Yn and has a nite (1− ε)-th moment for all ε > 0.

Now we apply this Lemma to the random variables T∞,(i)ext which are integrable underP and we get a certain random variable Tsup. We add to our probability spaces a copyof Tsup which is independent of all other random variables. Then for any t ≥ 0,

P[Zkn ≥ t] ≤ P[ C

E[G(p∞p2(h0n+k)/3)]

G(p∞p2(h0n+k)/3)∑

i=1

T∞,(i)exc ≥ t]

≤∑k≥0

P[G(p∞p2(h0n+k)/3) = k]P

[ C

E[G(p∞p2(h0n+k)/3)]

k∑i=1

T∞,(i)exc ≥ x]

≤∑k≥0

P[G(p∞p2(h0n+k)/3) = k]P

[C

k

E[G(p∞p2(h0n+k)/3)]

Tsup ≥ t]

≤ P[C

G(p∞p2(h0n+k)/3)

E[G(p∞p2(h0n+k)/3)]

Tsup ≥ t],

and since p∞p2(h0n+k)/3 < 1/3, we can use the fact that for any a < 1/3 we have

G(a)/E[G(a)] 3/2e. This shows that

for all n ≥ 0, and k > −n, Zkn CeTsup,

where e and Tsup are independent, so that the right-hand side has nite (1 − ε)-thmoment for all ε > 0. This nishes the proof of the second part in Proposition 9.1.

10 Sums of i.i.d. random variables

This section is completely self-contained and the notations used here are not relatedto those used previously.

Set β > 1 and let (Xi)i≥0 be a sequence of i.i.d. integer-valued non-negative randomvariables such that

P [X1 ≥ n] ∼ CXβ−γn, (10.1)

for CX ∈ (0,∞) and γ > 0.Let (X

(l)i )i≥0 be a sequence of i.i.d. integer-valued non negative random variables with

the law of Xi conditioned on Xi ≥ f(l), where f : N→ N is such that l − f(l)→∞.

80

10. SUMS OF I.I.D. RANDOM VARIABLES

Let (Z(l)i )i≥0,l≥0 be another sequence of i.i.d. non-negative random variables and let

Z(l),(k)i have the law of Z(l)

i under P [ · | X(l)i = l + k], if this last probability is well

dened, and as Z(l),(k)i = 0 otherwise. Dene

for k ∈ Z, l ≥ 0, F(l)

k (x) := P [Z(l),(k)i > x], (10.2)

We introduce the following assumptions.

1. There exists a certain random variable Z∞ such that

for all k ∈ Z and l ≥ 0, Z(l),(k)i

d−→ Z∞.

2. There exists a random variable Zsup such that

for all l ≥ 0, k ≥ −(l − f(l)) and i ≥ 0, Z(l),(k)i Zsup,

and E[Zγ+εsup ] <∞ for some ε > 0.

Moreover set

Y(l)i = Z

(l)i β

X(l)i and S(l)

n =n∑i=1

Y(l)i ,

and for λ ∈ [1, β), (λl)l≥0 converging to λ and l ∈ N, dene

N(λ)l =

⌊λγl β

γ(l−f(l))⌋,

K(λ)l = λβl.

Finally we denote by F∞(x) = P [Z∞ > x] the tail function of Z∞.

Theorem 10.1. Suppose that γ < 1 and Assumptions (1) and (2) hold true. Then wehave

for all λ ∈ [1, β) and (λl)l≥0 going to λ,S

(l)

N(λ)l

K(λ)l

→ I(dλ, 0,Lλ),

where I is an innitely divisible law. The Lévy spectral function Lλ satises

for all λ > 0 and x ∈ R, Lλ(x) = λγL1(λx) and Lβ(x) = L1(x), (10.3)

and

L1(x) =

0 if x < 0,

−(1− β−γ)∑k∈Z

βγkF∞(xβk) if x > 0. (10.4)

In particular, I(dλ, 0,Lλ) is continuous. Moreover, dλ is given by

dλ = λ1+γ(1− β−γ)∑k∈Z

β(1+γ)kE[ Z∞

(λβk)2 + Z2∞

]. (10.5)

81


The fact that the quantities appearing above are well dened will be treated in thecourse of the proof.

In order to prove Theorem 10.1 we will apply Theorem 4 in [10], which is itself aconsequence of Theorem IV.6 (p. 77) in [80].

Theorem 10.2. Let n(t) : [0,∞) → N and for each t let Yk(t) : 1 ≤ k ≤ n(t) be asequence of independent identically distributed random variables. Assume that for everyε > 0, it is true that

limt→∞

P [Y1(t) > ε] = 0. (10.6)

Now let L(x) : R \ 0 → R be a Lévy spectral function, d ∈ R and σ > 0. Then thefollowing statements are equivalent :

(i)

limt→∞

n(t)∑k=1

Yk(t)d−→ Xd,σ,L for t→∞

where Xd,σ,L has law I(d, σ,L).(ii) Dene for τ > 0 the random variable Zτ (t) := Y1(t)1|Y1(t)| ≤ τ. Then if x isa continuity point of L,

L(x) =

limt→∞ n(t)P [Y1(t) ≤ x] , for x < 0,− limt→∞ n(t)P [Y1(t) > x] , for x > 0,

σ2 = limτ→0

lim supt→∞

(n(t)Var(Zτ (t))) ,

and for any τ > 0 which is a continuity point of L(x),

d = limn→∞

n(t)E [Zτ (t)] +

∫|x|>τ

x

1 + x2dL(x)−

∫τ≥|x|>0

x3

1 + x2dL(x).

The condition (10.6) is veried in the course of the proof, in our context n(t) goesto innity.

10.1 Computation of the Lévy spectral function

Fix λ ∈ [1, β) and assume that x > 0 is a continuity point of Lλ. We want to showthat

− liml→∞

N(λ)l P

[Y

(l)1

K(λ)l

> x

]= Lλ(x). (10.7)

The discontinuity points of Lλ are exactly Cλ = (βkyn)/λ, k ∈ Z, n ∈ N whereyn, n ∈ N are the discontinuity points of F∞ (these sets are possibly empty).

82


Let us introducefor k ∈ Z, a

(l)k := P [X

(l)1 ≥ l + k]. (10.8)

Since N (λ)l ∼ (λβl−f(l))γ , we can write, recalling(10.2)

βγ(l−f(l))P( Y (l)

1

K(λ)l

> x)

=∑k∈Z

1k ≥ −(l − f(l))F (l)

k (λxβ−k)βγ(l−f(l))(a

(l)k − a(l)

k+1

).

Now recalling (10.1) and (10.8), we see that for l→∞,

βγ(l−f(l))a(l)k → β−γk . (10.9)

using l − f(l)→∞, the fact that λxβk is a continuity point of F∞ (because x > 0 is acontinuity point of Lλ) for any k and Assumption (1), we see that for all k ∈ Z

1k ≥ −(l − f(l))F (l)


(l)k − a(l)

k+1

)→F∞(λxβ−k)β−γk(1− β−γ) for l→∞ . (10.10)

In order to exchange limit and summation, we need to show that the terms of the sumare dominated by a function which does not depend on l and is summable. RecallingAssumption (2) and using (10.1) we see that βγ(l−f(l))a

(l)k ≤ C1β

−γk and∑k∈Z

1k ≥ −(l − f(l))F (l)


(l)k − a(l)

k+1

)≤C

∑k∈Z

F sup(λxβ−k)β−γk,

where F sup(x) = P [Zsup > x]. This last sum converges clearly for k →∞, and to showthat it converges for k → −∞ we simply notice that for any y > 0∑

k>0

E[1Zsup > yβk]βγk = E[ ∑

0<k≤bln(Zsup/y)/ lnβc

βγk]

≤ (1− β−γ)−1E[βγ ln(Zsup/y)/ lnβ] <∞,since we assume that E[Zγ+ε

sup ] <∞.Hence we can exchange limit and sum. Using (10.10) and the fact that l−f(l)→∞,

we get

− liml→∞

N(λ)l P

[Y

(l)1

K(λ)l

> x

]= −λγ(1− β−γ)

∑k∈Z

F∞(λxβk)βγk,

and, taking into account (10.3), this proves (10.7).

83


10.2 Computation of dλ

Fix λ ∈ [1, β). Since the integral∫ τ

0xdLλ is well dened, it suces to show that for

all τ ∈ Cλ, τ > 0

dλ = liml→∞

N(λ)l

K(λ)l

E[Y

(l)1 1Y (l)

1 < τK(λ)l ]−∫ τ

0

xdLλ +

∫ ∞0

x

1 + x2dLλ. (10.11)

First let us notice that N (λ)l /K

(λ)l ∼ (λβl)γ−1β−γf(l). We introduce

for all u > 0, G(l)k (u) = E

[Z

(l)1 1Z(l)

1 ≤ u|X(l)1 = k + l

]. (10.12)

Considering the rst term in (10.11), we compute

β(γ−1)l−γf(l)E[Y

(l)1 1Y (l)

1 < τλβl]

=∑k∈Z

1k ≥ −(l − f(l))[(a

(l)k − a(l)

k+1

)βγ(l−f(l))

]βkG

(l)k (τλβ−k).

Using l−f(l)→∞, (10.9) and Assumption (1), we see that for all k ∈ Z and τ ∈ Cλ,

1k ≥ −(l − f(l))[(a

(l)k − a(l)

k+1

)βγ(l−f(l))

]βkG

(l)k (τλβ−k) (10.13)

→ (1− β−γ)β(1−γ)kG∞(τλβ−k),

whereG∞(x) = E[Z∞1Z∞ ≤ x]. (10.14)

Once again we need to show that we can exchange limit and sum, which amounts tond a summable dominating function which does not depend on l. Using the fact thatfor u > 0

G(l)k (u) ≤ u and β−(γ−1)kG

(l)k (uβ−k) ≤ βεku1−γ−εE[Zγ+ε

sup ],

(to see the second inequality, use E[Y 1Y ≤ s] ≤ saE[Y 1−a1Y ≤ s] with a = 1 −γ − ε)), we get that∑

k∈Z

1k ≥ −(l − f(l))[(a

(l)k − a(l)

k+1

)βγ(l−f(l))

]βkG

(l)k (τλβ−k)

≤C(τλ∑k≥0

β−γk + (τλ)1−γ−εE[Zγ+εsup ]

∑k<0

βεk)<∞,

due to Assumption (2). Hence recalling (10.13), we get that for τ ∈ Cλ

liml→∞

N(λ)l

K(λ)l

E[Y

(l)1 1Y (l)

1 < τK(λ)l ]

= λγ−1(1− β−γ)∑k∈Z

βk(γ−1)G∞(τλβk). (10.15)

84


Furthermore, recalling (10.3) and (10.4), we get for τ ∈ Cλ∫ τ

0

xdLλ = λγ(1− β−γ)∫x≤τ

x∑k∈Z

βγkd(−F∞)(λxβk)

= λγ−1(1− β−γ)∑k∈Z

β(γ−1)k

∫λxβk≤λτβk

λxβkd(−F∞)(λxβk)

= λγ−1(1− β−γ)∑k∈Z

β(γ−1)kG∞(τλβk),

and this term exactly compensates for (10.15). Hence, we are left to compute in a similarfashion,

dλ =

∫ ∞0

x

1 + x2dLλ

=λγ(1− β−γ)∫ ∞

0

x

1 + x2

∑k∈Z

βγkd(−F∞)(λxβk)

=λ1+γ(1− β−γ)∑k∈Z

β(1+γ)k

∫ ∞0

λxβk

(λβk)2 + (λxβk)2d(−F∞)(λxβk)

=λ1+γ(1− β−γ)∑k∈Z

β(1+γ)kE[ Z∞

(λβk)2 + Z2∞

].

This sum is nite since the terms in the sum can be bounded from above byC1(λ)β−εkE[Zγ+ε

sup ] and C2(λ)βγk, where C1(λ) = maxx≥0 (x1−γ/(λ2 + x2)) and C2(λ) =maxx≥0 x/(λ

2 + x2). The rst upper bound is summable for k → ∞, the other fork → −∞ and so dλ is well-dened.

10.3 Computation of the variance

We show that for any λ ∈ [1, β) we have

σ2 = limτ→0

lim supl→∞

N(λ)l

(K(λ)l )2

Var(Y

(l)1 1Y (l)

1 ≤ τK(λ)l )

= 0. (10.16)

First, using (10.15), let us notice that

liml→∞

N(λ)l

(K(λ)l )2

E[Y

(l)1 1Y (l)

1 ≤ τK(λ)l ]

= 0. (10.17)

Further, we have N (λ)l /(K

(λ)l )2 ∼ (λβl)γ−2β−γf(l). Dene

for all u ≥ 0, H(l)k (u) = E

[(Z

(l)1

)2

1Z(l)1 ≤ u | X(l)

1 = k + l].

85


We compute

β(γ−2)l−γf(l)E[(Y

(l)1

)2

1Y (l)1 < τλβl

]=∑k∈Z

1k ≥ −(l − f(l))β(γ−2)l−γf(l)β2(k+l)H(l)k (τλβ−k)

(a

(l)k − a(l)

k+1

).

By (10.1) we have a(l)k β

γ(l−f(l)) ≤ C1β−γk, hence the terms of our sum are bounded

above by C1β(2−γ)kH

(l)k (τλβ−k). Note that H(l)

k (u) ≤ u2, so that

β(2−γ)kH(l)k (τλβ−k) ≤ β−γk(τλ)2,

which gives an upper bound for k ≥ 0. On the other hand, Assumption (2)

β(2−γ)kH(l)k (τλβ−k) ≤ βεk(τλ)2−γ−εE[Zγ+ε

sup ].

These inequalities imply that

lim supl→∞

N(λ)l

(K(λ)l )2

Var(Y (l)1 1Y (l)

1 ≤ τK(λ)l ) ≤ C2τ

2−γ−ε,

where C2 is nite and depends on ε and λ. Hence letting τ go to 0 yields the result,since in Assumption (2) we can assume ε to be as small as we need in particular it canbe chosen such that 2− γ − ε > 0.

11 Limit theorems

11.1 Proof of Theorem 1.3

Assume ε < min(1/4, 2γ/3). For λ > 0, we will study the limit distributions of thehitting time properly renormalized along the subsequences dened as follows

for k ∈ N, nλ(k) = bλf ′(q)−kc.First, recalling (9.3), using Proposition 9.1 and Lemma 3.3, we can apply Theo-

rem 10.1 to get

for any (λl)l≥0 going to λ,1

λβk

bλγnλ(k)

βγ(k−f(k))c∑i=1

χ∗i (nλ(k))d−→ Ydλ,0,Lλ , (11.1)

where f(k) := hnλ(k) = d−(1 − ε) ln(nλ(k))/ ln f ′(q)e and Ydλ,0,Lλ is a random vari-able whose law I(dλ, 0,Lλ) is the innitely divisible law characterized by (10.3), (10.3)and (10.5), where Z∞ is given by (1.3).

86

11. LIMIT THEOREMS

Using Proposition 8.1, (11.1) still holds if we replace χ∗i (n) by χi(n).Recalling Proposition 7.1 we have

b(1+o1(1))λρCaf ′(q)−(k−hnλ(k))c∑

i=1

χi(nλ(k)) χnλ(k) b(1+o2(1))λρCaf ′(q)

−(k−hnλ(k))c∑i=1

χi(nλ(k)),

where

1 + o1(1) = (1− n−ε/4)ρnρ

n

λf ′(q)−kf ′(q)henf ′(q)hn

and 1 + o2(1) = (1 + 2n−ε/4)ρnρ

n

λf ′(q)−k,

writing n for nλ(k) = bλf ′(q)−kc and n = n− (−2 lnn/ ln f ′(q)).Hence both sides of the previous equation, properly renormalized, converge in dis-

tribution to the same limit law, implying that (the law of) χ(n) converges to the samelaw as well. Recalling (5.3), this yields for any λ > 0

χ(nλ(k))

(ρCanλ(k))1/γ

d−→ Yd(ρCaλ)1/γ

,0,L(ρCaλ)1/γ

,

where Yd(ρCaλ)1/γ

,0,L(ρCaλ)1/γ

is a random variable with law I(d(ρCaλ)1/γ , 0,L(ρCaλ)1/γ ) and

we used that βγ = 1/f ′(q).Then by Proposition 5.1, we get that

∆nλ(k)

(ρCanλ(k))1/γ

d−→ Yd(ρCaλ)1/γ

,0,L(ρCaλ)1/γ

,

which proves Theorem 1.3.We note for further reference that

I(dλ, 0,Lλ) is continuous (11.2)

This follows from Theorem III.2 (p. 43) in [80], since, due to (10.4), limx→0L1(x) = −∞.


In order to prove Theorem 1.2, assume that(∆n/n

1/γ)n≥0 converges in law. It follows that all subsequential limits are the same, sothat

for all λ ∈ [1, β) and x ∈ R+, L1(x) = λγL1(λx).

Plugging in the values λ = β1/3 and x = β−2/3 gives∑k∈Z

f ′(q)−kP [Z∞ > βk−2/3] = f ′(q)−1/3∑k∈Z

f ′(q)−kP [Z∞ > βk−1/3]. (11.3)

87


We will show that for β → ∞, the right hand side and the left hand side of (11.3)have dierent limits. First for k ≥ 1, using Remark 6.1, we see that

P[Z∞ > βk−1/3] ≤ β1/3−kE[Z∞] ≤ β1/3−kE[S∞]E[G(p∞/3)] = β−(k−1)O(β−1/3), (11.4)

for β → ∞ where O(β−1/3) = β−2/3E[S∞]E[G(p∞/3)] does not depend on k (recallProposition 3.1 to see that E[S∞] is bounded in β) . In the same way,

P[Z∞ > βk−2/3] = β−(k−1)O(β−1/3), (11.5)

for O(·) independent of k ≥ 1.Hence,

limβ→∞

∞∑k=1

f ′(q)−kP [Z∞ > βk−2/3] = 0 = limβ→∞

∞∑k=1

f ′(q)−kP [Z∞ > βk−1/3]. (11.6)

For k ≤ 0, we have

P[Z∞ > βk−1/3] ≤ P[Z∞ > 0] ≤ P[Bin(W∞, p∞) > 0],

further, since S∞ ≥ 1, we see that

on Z∞ > 0, Z∞ ≥ S∞e1 ≥ e1,

where e1 is independent of the event Z∞ > 0 = Bin(W∞, p∞) > 0. Hence

P[Z∞ > βk−1/3] = 1− P[Z∞ ≤ βk−1/3]

≥ 1− P[Bin(W∞, p∞) = 0]− P [e1 ≤ β−1/3]

= P[Bin(W∞, p∞) > 0] + o(1),

for β →∞ and hence

P[Z∞ > βk−1/3] = P[Bin(W∞, p∞) > 0] + o(1), (11.7)

where o(·) does not depend on k.In the same way,

P[Z∞ > βk−2/3] = P[Bin(W∞, p∞) > 0] + o(1) (11.8)

for β →∞. Plugging (11.7) and (11.8) in equation (11.3) and taking into account (11.6)we see that

limβ→∞

∑k∈Z

f ′(q)−kP [Z∞ > βk−2/3] = limβ→∞

1

1− f ′(q)P [Bin(W∞, p∞) > 0]

88

11. LIMIT THEOREMS

and

limβ→∞

∑k∈Z

f ′(q)−kP [Z∞ > βk−1/3] = limβ→∞

1

1− f ′(q)P [Bin(W∞, p∞) > 0] .

Hence, we would have

limβ→∞

1

1− f ′(q)P [Bin(W∞, p∞) > 0] = lim

β→∞f ′(q)−1/3 1

1− f ′(q)P [Bin(W∞, p∞) > 0] .

This could only be possible if P [Bin(W∞, p∞) > 0]→ 0 for β →∞, but we know that

P [Bin(W∞, p∞) > 0] > p∞P[W∞ ≥ 1] > c > 0,

where c does not depend on β, see Lemma 6.6. This proves Theorem 1.2.In particular, if β is large enough, I(d1, 0,L1) is not a stable law and this implies

(vii) in Theorem 1.4.


We will show that

limM→∞

lim supn→∞

P[ ∆n

n1/γ/∈ [1/M,M ]

]= 0. (11.9)

This implies in particular that the family (∆n/n1/γ)n≥0 is tight. We will then prove

limM→∞

lim supn→∞

P[ |Xn|nγ

/∈ [1/M,M ]]

= 0, (11.10)

which implies that the family (|Xn|/nγ)n≥0 is tight. (1.2) is then a consequence of (11.9)and (11.10).

To show (11.9), note that for n ∈ [f ′(q)−k, f ′(q)−(k+1))

P[

∆n

n1/γ/∈ [1/M,M ]

]≤P[

∆f ′(q)−k

(f ′(q)−kρCa)1/γ<

M

(f ′(q)ρCa)1/γ

]+ P

[∆f ′(q)−(k+1)

(f ′(q)k+1ρCa)1/γ>

1

M(f ′(q)−1ρCa)1/γ]

].

Using Theorem 1.3 we get

lim supn

P[

∆n

n1/γ/∈ [1/M,M ]

]

≤ P[Yd

(ρCa)1/γ,0,L

(ρCa)1/γ/∈[

1

M(f ′(q)−1ρCa)1/γ,

M

(f ′(q)ρCa)1/γ

]].

89


where Yd(ρCa)1/γ

,0,L(ρCa)1/γ

is a random variable with law I(d(ρCa)1/γ , 0,L(ρCa)1/γ ). Here we

used that the limiting law I(dx, 0,Lx) is continuous, see (11.2), and has in particular noatom at 0, so we get that

limM→∞

P [Ydx,0,Lx /∈ [1/M,M ]] = 0,

which proves (11.9).Let us prove (11.10). Let n ≥ 0 and write nγ = λ0f

′(q)−i0 for some i0 ∈ N andλ0 ∈ [1, 1/f ′(q)). Let i ∈ N. To control the probability that |Xn| is be much larger thannγ, note that

P[ |Xn|nγ≥ λ−1

0 f ′(q)−i]≤ P[∆b(λ−1

0 f ′(q)−i)(λ0f ′(q)−i0 )c < (λ0f′(q)−i0)1/γ]

= P[ ∆bf ′(q)−i−i0c

(ρCaf ′(q)−i−i0)1/γ< (λ0ρCaf

′(q)−i)−1/γ].

Hence for any ε > 0, and i large enough such that (ρCaf′(q)−i)−1/γ < ε,

P[ |Xn|nγ≥ f ′(q)−i−1

]≤ P

[ |Xn|nγ≥ λ−1

0 f ′(q)−i]≤ P

[ ∆bf ′(q)−i−i0c(ρCaf ′(q)−i−i0)1/γ

< ε].

Now, using Theorem 1.3, taking n (i.e. i0) to innity, we get that for any ε > 0,

for i large enough, lim supn

P[ |Xn|nγ≥ f ′(q)−i−1

]≤ P [Yd1,0,L1 ≤ ε],

using (11.2) and hence

lim supM→∞

lim supn→∞

P[ |Xn|nγ≥M

]≤ lim sup

ε→0P [Yd1,0,L1 ≤ ε] = 0 . (11.11)

Next, we will consider the probability that |Xn| is much smaller than nγ. Let usdenote

Back(n) = maxi<j≤n

(|Xi| − |Xj|) ,

the maximal backtracking of the random walk. It is easy to see that

Back(n) ≤ max2≤i≤n

(τi − τi−1) ∨ τ1 .

Hence since τ1 and τ2 − τ1 have exponential moments

P[Back(n) ≥ nγ/2

] ≤ Cn exp(−cnγ/2). (11.12)

90

11. LIMIT THEOREMS

If the walk is at a level inferior to (1/M)nγ at time n and has not backtracked morethan nγ/2, it has not reached (2/M)nγ. This implies that for all M > 0,

P[ |Xn|nγ

< 1/M

]≤ P[Back(n) ≥ nγ/2] + P

[∆b(2/M)nγc

(ρCa2/M)1/γn> (ρCa2/M)1/γ

].

Hence, using a reasoning similar to the proof of (11.11), we have

limM→∞

lim supn→∞

P[ |Xn|nγ

< 1/M

]≤ lim inf

M→∞P [Yd1,0,L1 ≥M ] = 0. (11.13)

Using (11.11) and (11.13), we get

limM→∞

lim supn→∞

P[ |Xn|nγ

/∈ [1/M,M ]

]= 0, (11.14)

which shows (11.10) in Theorem 1.1.Let us prove (iii) in Theorem 1.1. We have

P[

limn→∞

ln |Xn|lnn

6= γ

]≤ P

[lim supn→∞

ln |Xn|lnn

> γ

]+ lim

M→∞P[lim infn→∞

|Xn|nγ≤ 1

M

]Using Fatou's Lemma,

P[lim infn→∞

|Xn|nγ

<1

M

]≤ lim inf

n→∞P[ |Xn|nγ

< 1/M

],

and taking M to innity we get

P[

limn→∞

ln |Xn|lnn

6= γ

]≤ P

[lim supn→∞

ln |Xn|lnn

> γ

].

Set ε > 0, we have

P[lim supn→∞

ln |Xn|lnn

> (1 + 2ε)γ

]≤ P

[lim supn→∞

|Xn|n(1+ε)γ

≥ 1

]≤ P

[lim supn→∞

supi≤n |Xi|n(1+ε)γ

≥ 1

].

Dene

D′(n) =

max` ∈ ∪i=0,...,∆Y

nLYi

H(`) ≤ 4 lnn

− ln f ′(q)

.

Denoting t(n) such that στt(n)≤ n < στt(n)+1

, we have

for ω ∈ D′(n), |Xστt(n)| ≤ |Xn| ≤ |Xστt(n)+1

|+ 4 lnn

− ln f ′(q), (11.15)

91


and since using B1(n) dened right above Lemma 7.1, we get

for ω ∈ B1(n), |Xστt(n)+1| ≤ |Xστt(n)

|+ nε. (11.16)

We have using Lemma 5.1 and (3.7)

P[D′(n)c] ≤ P[A1(n)c] + P[A1(n), card∪∆Y

ni=1LYi > n2

]+ P

[card∪∆Y

ni=1LYi ≤ n2, D′(n)c

]≤ O(n−2) + P

[C1n∑i=0

cardL(i)0 > n2

]+ n2Q

[H ≥ 4 lnn

− ln f ′(q)

]≤ O(n−2) + n−4Var

(C1n∑i=0

cardL(i)0

)+ n2n−4 = O(n−2),

where we used that cardL(i)0 are i.i.d. random variables which are L2 since they are

stochastically dominated by the number of ospring Z which is L2 by our assumption.By Lemma 7.1, the previous estimate and Borel-Cantelli we have ω ∈ B1(n)∩D(n)

asymptotically, we get recalling (11.15) and (11.16) that for ε < γ

P[lim supn→∞


≥ 1

]≤ P

[lim infn→∞

( |Xστt(n)|

n(1+ε)γ+ o(1)

)≥ 1

].

Since |Xστt(n)| ≤ |Xn| we have

P[

limn→∞


≥ 1

]≤ P

[lim infn→∞

|Xστt(n)|

n(1+ε)γ≥ 1

]

≤ lim infn→∞

P[ |Xn|n(1+ε)γ

≥ 1

]≤ lim inf

M→∞lim infn→∞

P[ |Xn|nγ≥M

]= 0,

where we used Fatou's Lemma and (ii) in Theorem 1.1.Now since this result is true for all ε > 0 small enough we get

P[

limn→∞

ln |Xn|lnn

6= γ

]≤P[lim supn→∞

ln |Xn|lnn

> γ

]≤ lim inf

ε→0P[lim supn→∞

ln |Xn|lnn

> (1 + 2ε)γ

]= 0,

which nishes the proof of (1.2).

92

11. LIMIT THEOREMS


It remains to show (iv), (v), (vi) and (viii) in Theorem 1.4.

Démonstration. We start by proving (viii). Recall

Z∞ =S∞

1− β−1


ei,

and in particular the fact that S∞, W∞ and the i.i.d. exponential random variablesei are independent. Let S∞ = S∞/p∞ and denote its law by ν∞. Further, let αk =

P[Bin(W∞, p∞) = k], k = 0, 1, 2, . . .. Conditioned on S∞ and Bin(W∞, p∞), the law ofZ∞ is a Gamma distribution. More precisely, for any test function ϕ,

E[ϕ(Z∞)] = α0ϕ(0) +∞∑k=1

∞∫0

∞∫0

ϕ(su)e−uuk−1

(k − 1)!du

ν∞(ds)αk

= α0ϕ(0) +∞∑k=1

∞∫0

∞∫0

ϕ(v)e−v/svk−1

(k − 1)!

1

skdv

ν∞(ds)αk

= α0ϕ(0) +

∞∫0

ϕ(v)∞∑k=1

αkvk−1

(k − 1)!EQ

[e−v/

eS∞ (S∞)−k] dvWe point out that, due to Lemma 6.6, we have 0 < α0 < 1. Hence, Z∞ has an atom

of mass α0 at 0 and the conditioned law of Z∞, conditioned on Z∞ > 0, has the densityψ, where

ψ(v) =∞∑k=1

αkvk−1

(k − 1)!EQ

[e−v/

eS∞ (S∞)−k]

= EQ

[1

S∞e−v/

eS∞ ∞∑k=1

αk(k − 1)!

(v

S∞

)k−1]

Using the fact that S∞ ≥ 2 and lim sup 1k

logαk < 0 (see Lemma 6.2), we see that ψis bounded and C∞. Note that since S∞ and W∞ have nite expectation, Z∞ has alsonite expectation and in particular

∞∫0

vψ(v)dv <∞ . (11.17)

93


This shows (viii) in Theorem 1.4. We will later need that

∞∫0

v1+γ|ψ′(v)|dv <∞ . (11.18)

To show (11.18), note that ψ′(v) equals

EQ

[−1

(S∞)2e−v/

eS∞ ∞∑k=1

αk(k − 1)!

(v

S∞

)k−1

+1

(S∞)2e−v/

eS∞ ∞∑k=2

αk(k − 2)!

(v

S∞

)k−2]

which implies, with α := lim sup(α

1/kk

)< 1,

|ψ′(v)| ≤ C1EQ

[1

(S∞)2e−(1−α)v/eS∞

].

But, for δ ∈ (0, 1),

EQ

[1

(S∞)2e−(1−α)v/eS∞

]

≤ EQ

[e−(1−α)v/eS∞1S∞ < v1−δ

]+ EQ

[1

(S∞)21S∞ ≥ v1−δ

]≤ e−(1−α)vδ +

1

v3−3δEQ

[S∞

]Now, choosing δ small enough such that 3δ + γ < 1 yields (11.18).

We next show that the function L1 is absolutely continuous. Recalling (i) in Theorem1.4 we see that, for x > 0,

−(1− β−γ)−1L1(x) =∑k∈Z

βγkF∞(xβk)

=∑k∈Z

βγk∞∫

xβk

ψ(v)dv

=

∞∫0

(∑k∈Z

βγk1v ≥ xβk)ψ(v)dv .

Now, ∑k∈Z

βγk1v ≥ xβk =∑

k≤K(v/x)

βγk =: g(vx

)94

11. LIMIT THEOREMS

where, setting u = vx, K(u) = b log u

log βc. An easy computation gives

g(u) =βγ(K(u)+1)

βγ − 1. (11.19)

Hence, for x > 0,

−(1− β−γ)−1L1(x) =

∞∫0

g(vx

)ψ(v)dv = x ·

∞∫0

g(u)ψ(xu)du . (11.20)

The last formula shows, noting that g(u) is of order uγ for u→∞ and recalling (11.17)and (11.18), that L1 is C1 and in particular absolutely continuous. Due to the scalingrelation (ii), the same holds true for Lλ. This shows (iv) in Theorem 1.4.

Due to (11.19), we have

1

βγ − 1uγ ≤ g(u) ≤ βγ

βγ − 1uγ

Plugging this into the rst equality in (11.20) yields (1.4). This proves (v) in Theorem1.4. To show (vi), we use a result of [103] which says that an innite divisible law isabsolutely continuous if the absolutely continuous component Lac of its Lévy spectral

function satises∞∫−∞

dLac(x) =∞, see also [80], p. 37. In our case, this is satised since

Lac1 (x) = L1(x) and limx→0L1(x) = −∞. Further, the statement about the moments of µλ

follows from the corresponding statement about the moments of Lλ, see [84] or [80], p.36.

Acknowledgements

A.F. would like to thank his advisor Christophe Sabot for his support and manyuseful discussions.

We also thank the Institut Camille Jordan, Université de Lyon 1 the Institut fürMathematische Statistik, Universität Münsterand Courant Institute of MathematicalSciences, New York University for giving A.F. the possibility to visit Münster and A.F.and N.G. the possibility to visit New York.

95


96

5The speed of a biased random walk on

a percolation cluster at high density

We study the speed of a biased random walk on a percolation cluster on Zd infunction of the percolation parameter p. We obtain a rst order expansion of the speedat p = 1 which proves that percolating slows down the random walk at least in the casewhere the drift is along a component of the lattice.

The material of this chapter has been submitted for publication, see [38].

1 Introduction

Random walks in reversible random environments are an important subeld of ran-dom walks in random media. In the last few years a lot of work has been done to un-derstand these models on Zd, one of the most challenging being the model of reversiblerandom walks on percolation clusters, which has raised many questions.

In this model, the walker is restrained to a locally inhomogeneous graph, makingit dicult to transfer any method used for elliptic random walks in random media. In

CHAPITRE 5. THE SPEED OF A BIASED RANDOM WALK ON APERCOLATION CLUSTER AT HIGH DENSITY

the beginning results concerned simple random walks, the question of recurrence andtransience (see [48]) was rst solved and latter on a quenched invariance principles wasproved in [13] and [72]. More recently new results (e.g. [71] and [14]) appeared, but stillunder the assumption that the walker has no global drift.

The case of drifted random walks on percolation cluster features a very interestingphenomenon which was rst described in the theoretical physics literature (see [28]and [29]), as the drift increases the model switches from a ballistic to a sub-ballisticregime. From a mathematical point of view, this conjecture was partially addressedin [15] and [99]. This slowdown is due to the fact that the percolation cluster containsarbitrarily large parts of the environment which act as traps for a biased random walk.This phenomenon, and more, is known to happen on inhomogeneous Galton-Watsontrees, cf. [68], [9] and [8].

Nevertheless this model is still not well understood and many questions remain open,the most famous being the existence and the value of a critical drift for the expectedphase transition. Another question of interest is the dependence of the limiting velocitywith respect to the parameters of the problem i.e. the percolation parameter and thebias. This last question is not specic to this model, but understanding in a quantitative,or even qualitative way, the behaviour of speed of random walks in random media seemsto be a dicult problem and very few results are currently available on Zd (see [90]).

In this article we study the dependence of the limiting velocity with respect to thepercolation parameter around p = 1. We try to adapt the methods used in [90] whichwere introduced to study environments subject to small perturbations in a uniformly-elliptic setting. For biased-random walk on a percolation cluster of high density, the walkis subject to rare but arbitrarily big pertubations so that the problem is very dierentand appears to be more dicult.

The methods rely mainly on a careful study of Kalikow's auxilary random walk whichis known to be linked to the random walks in random environments (see [104] and [100])and also to the limiting velocity of such walks when it exists (see [90]). Our main taskis to show that the unbounded eects of the removal of edges, once averaged over allcongurations, is small. This will enable us to consider Kalikow's auxilary random walkas a small perturbation of the biased random walk on Zd. As far as we know it is the rsttime such methods are used to study a random conductance model or even a non-ellipticrandom walks in random media.

2 The model

The models presented in [15] and [99] are slightly dierent, we choose to considerthe second one as it is a bit more general, since it allows the drift to be in any direc-tion. Nevertheless all the following can be adapted without any diculty to the modeldescribed in [15].

98

2. THE MODEL

Let us describe the environment, we consider the set of edges E(Zd) of the latticeZd for some d ≥ 2. We x p ∈ (0, 1) and perform a Bernoulli bond-percolation, that iswe pick a random conguration ω ∈ Ω := 0, 1E(Zd) where each edge has probabilityp (resp. 1 − p) of being open (resp. closed) independently of all other edges. Let usintroduce the corresponding measure

Pp = (pδ1 + (1− p)δ0)⊗E(Zd).

Hence an edge e will be called open (resp. closed) in the conguration ω if ω(e) = 1(resp. ω(e) = 0). This naturally induces a subgraph of Zd which will be denoted ω andit also yields a partition of Zd into open clusters.

It is classical in percolation that for p > pc(d), where pc(d) ∈ (0, 1) denotes thecritical percolation probability of Zd (see [46]), we have a unique innite open clusterK∞(ω), Pp-a.s.. Moreover the following event has positive Pp-probability

I = there is a unique innite cluster K∞(ω) and it contains 0.

In order to dene the random walk, we introduce a bias ` = λ~ of strengh λ > 0 anda direction ~ which is in the unit sphere with respect to the Euclidian metric of Rd. Ona conguration ω ∈ Ω, we consider the Markov chain of law P ω

x on Zd with transitionprobabilities pω(x, y) for x, y ∈ Zd dened by

1. X0 = x, P ωx -a.s.,

2. pω(x, x) = 1, if x has no neighbour in ω,

3. pω(x, y) =cω(x, y)∑z∼x c

ω(x, z),

where x ∼ y means that x and y are adjacent in Zd and also we set

for all x, y ∈ Zd, cω(x, y) =

e(y+x)·` if x ∼ y and ω(x, y) = 1,

0 otherwise.

We see that this Markov chain is reversible with invariant measure given by

πω(x) =∑y∼x

cω(x, y).

Let us call cω(x, y) the conductance between x and y in the conguration ω, thisis natural because of the links existing between reversible Markov chains and electricalnetworks. We will be making extensive use of this relation and we refer the reader to [30]and [70] for a further background. Moreover for an edge e = [x, y] ∈ E(Zd), we denotecω(e) = cω(x, y) and rω(e) = 1/cω(e).

Finally the annealed law of the biased random walk on the innite percolation clusterwill be the semi-direct product Pp = Pp[ · | I ]× P ω

0 [ · ].

99


The starting point of our work is the existence of a constant limiting velocity whichwas proved in [99] and with some additional work sznitman-perco managed to obtainthe following result

Theorem 2.1. For any d ≥ 2, p ∈ (pc(d), 1) and any ` ∈ Rd∗, there exists v`(p) ∈ Rd

such that

for ω −Pp[ · | I ]− a.s., limn→∞

Xn

n= v`(p), P

ω0 − a.s..

Moreover there exist λ1(p, d, `), λ2(p, d, `) ∈ R+ such that

1. for λ = ` · ~ < λ1(p, d, `), we have v`(p) · ~ > 0,

2. for λ = ` · ~ > λ2(p, d, `), we have v`(p) = 0.

Our main result is a rst order expansion of the limiting velocity with respect tothe percolation parameter at p = 1. As in [90], the result depends on certain Greenfunctions dened for a conguration ω as

for any x, y ∈ Zd, Gω(x, y) := Eωx

[∑n≥0

1Xn ∈ y].

Before stating our main theorem we recall that v`(1) =∑

e∈ν p(e)e, where ω0 is theenvironment at p = 1, p(e) = pω0(0, e), and ν is the set of unit vectors of Zd.

Theorem 2.2. For d ≥ 2, p ∈ (pc(d), 1) and for any ` ∈ Rd∗, we have

v`(1− ε) = v`(1)− ε∑e∈ν

(v`(1) · e)(Gωe0(0, 0)−Gωe0(e, 0))(v`(1)− de) + o(ε),

where for any e ∈ ν we denote

for f ∈ E(Zd), ωe0(f) = 1f 6= e and de =∑e′∈ν

pωe0(0, e′)e′,

are respectively the environment where only the edge [0, e] is closed and its correspondingmean drift at 0.

Proposition 2.1. Let us denote Je = Gω0(0, 0) − Gω0(e, 0) for e ∈ ν. We can rewritethe rst term of the expansion in the following way

v′`(1) =∑e∈ν

(v`(1) · e) p(e)Je

1− p(e)Je − p(−e)J−e (e− v`(1)),

so that if for e ∈ ν such that v`(1) · e > 0 we have v`(1) · e ≥ ||v`(1)||22 then

v`(1) · v′`(1) > 0,

which in words means that the percolating slows down the random walk at least at p = 1.The previous condition is veried for example in the following cases

100

2. THE MODEL

1. ~ ∈ ν, i.e. when the drift is along a component of the lattice,

2. ` = λ~, where λ < λc(~) for some λc(~) > 0, i.e. when the drift is weak.

Remark 2.1. The property of Proposition 2.1 is expected to hold for any drift, but wewere unable to carry our the computations. More generaly the previous should be true ina great variety of cases, in particular one could hope it holds in the whole supercriticalregime. For a somewhat related conjecture, see [18].

Remark 2.2. Another natural consequence which is not completely obvious to prove isthat the speed is positive for p close enough to 1.

Remark 2.3. Finally, this result can give some insight on the dependence of the speedwith respect to the bias. Indeed, x a bias ` and some µ > 1, Theorem 2.2 implies thatfor ε0 = ε0(`, µ) > 0 small enough we have

vµ`(1− ε) · ~ > v`(1− ε) · ~ for ε < ε0.

Before turning to the proof of this result, we introduce some further notations. Letus also point out that we will refer to the percolation parameter as 1 − ε instead of pand assume ε < 1/2. In particular we have 1− ε > pc(d) for all d ≥ 2.

We denote by x ↔ y the event that x and y are connected in ω. If we want toemphasize the conguration we will use x ω↔ y. Accordingly, let us denote Kω(x) thecluster (or connected component) of x in ω.

Given a set V of vertices of Zd we denote by |V | its cardinality, by E(V ) = [x, y] ∈E(Zd) | x, y ∈ V its edges and

∂V = x ∈ V | y ∈ Zd \ V, x ∼ y, ∂EV = [x, y] ∈ E(Zd) | x ∈ V, y /∈ V ,

and also for B a set of edges of E(Zd) we denote

∂B = x | ∃ y, z, [x, y] ∈ B, [x, z] /∈ B, ∂EB = [x, y] | x ∈ ∂B, y /∈ ∂B ∪ V (B),

where V (B) = x ∈ Zd | ∃y ∈ Zd [x, y] ∈ B.Given a subgraph G of Zd containing all vertices of Zd, we denote dG(x, y) the graph

distance in G induced by Zd between x and y, moreover if x and y are not connected inG we set dG(x, y) = ∞. In particular dω(x, y) is the distance in the percolation clusterif x↔ y. Moreover for x ∈ G and k ∈ N, we denote the ball of radius k by

BG(x, k) = y ∈ G, dG(x, y) ≤ k and BEG(x, k) = E(BG(x, k)),

where we will omit the subscript when G = Zd.Let us denote by (e(i))i=1...d an orthonormal basis of Zd such that e(1) · ~≥ e(2) · ~≥

· · · ≥ e(d) · ~≥ 0, in particular we have e(1) · ~≥ 1/√d.

101


In order to control volume growth let us dene ρd such that

for all r ≥ 1, |B(x, r)| ≤ ρdrd and |∂B(x, r)| ≤ ρdr

d−1.

We will need to modify the conguration of the percolation cluster at certain vertices.So given A1, A2 ∈ E(Zd), B1 ∈ 0, 1A1 and B2 ∈ 0, 1A2 , let us denote ωA1,B1

A2,B2the

conguration such that

1. ωA1,B1

A2,B2([x, y]) = ω([x, y]), if [x, y] /∈ A1 ∪ A2,

2. ωA1,B1

A2,B2([x, y]) = 1[x, y] /∈ B1, if [x, y] ∈ A1,

3. ωA1,B1

A2,B2([x, y]) = 1[x, y] /∈ B2, if [x, y] ∈ A2 \ A1,

which in words means that we impose that the closed edges of A1 (resp. A2) are exactlythose of B1 (resp. B2), and in case of an intersection between A1 and A2 the conditionimposed by A1 is most important. For k1, k2 ≥ 1, z1, z2 ∈ Zd and B1, B2 ∈ 0, 1BE(z1,k)×0, 1BE(z2,k), we introduce

ω(z1,k1),B1

(z2,k1),B2:= ω

BE(z1,k1),B1

BE(z2,k2),B2and for B1, B2 ∈ 0, 1ν , ωz1,B1

z2,B2:= ω

(z1,1),z+B1

(z2,1),z+B2, (2.1)

to describe congurations modied on balls. Moreover

ωA,1 := ωA,∅ and ωA,0 := ωA,A, (2.2)

to denote in particular the special cases where all (resp. no) edges of A are open. Wewill use of combinations of these notations, for example, ω(z,k),1 := ωB

E(z,k),∅.In connection with that, for a given conguration ω ∈ Ω, we call conguration of z

and denoteC(z) = e ∈ ν, ω([z, z + e]) = 0,

the set of closed edges adjacent to z.Hence we can denote e ∈ ν and A ∈ 0, 1ν

pA(e) = pω0,ν\A

(0, e), c(e) = cω0,1

(e) and πA = πω0,ν\A

(0). (2.3)

Furthermore the pseudo elliptic constant κ0 = κ0(`, d) > 0 will denote

κ0 = minA∈0,1ν ,e/∈A

pA(e), (2.4)

which is the minimal non-zero transition probability.Similarly we x κ1 = κ1(`, d) > 0 such that

1

κ1

πωz,A

(z) ≤ e2λz.~ ≤ κ1πωz,A(z), (2.5)

for any A ∈ 0, 1ν \ ν and z ∈ Zd.

102

3. KALIKOW'S AUXILIARY RANDOM WALK

Finally τδ will denote a geometric random variable of parameter 1 − δ independentof the random walk and the environment moreover for A ⊂ Zd set

TA = infn ≥ 0, Xn ∈ A and T+A = infn ≥ 1, Xn ∈ A,

and for z ∈ Zd we denote Tz (resp. T+z ) for Tz (resp. T

+z).

Concerning constants we choose to denote them by Ci for global constants, or γi forlocal constants and will implicitely be supposed to be in (0,∞). Their dependence withrespect to d and ` will not always be specied.

Let us present the structure of the paper. In Section 3, we will introduce the centraltool for the computation of the expansion of the speed : Kalikow's environment andlink it to the asymptotic speed. Then, we will concentrate on getting the continuityof the speed, mathematically the problem is simply reduced to giving upper boundson quantities depending on Green functions, on a more heuristical level our aim is tounderstand the slowdown induced by unlikely congurations where trapsappear. Sincegetting the upper bound is a rather complicated and technical matter we will rst give aquick sketch, as soon as further notations are in place, and try to motivate our approachat the end of the next section.

In Section 4 and Section 5, we will respectively give estimates on the behaviour ofthe random walk near traps and on the probability of appearence of such traps in thepercolation cluster. Then in Section 6 we will put together the previous results to provethe continuity of the speed.

The proof of Theorem 2.2 will be done in Section 7. In order to obtain the rstorder expansion, the task is essentially similar to obtaining the continuity, but thecomputations are much more involved and will partly be postponed to Section 8.

Finally Proposition 2.1 is proved in Section 9.

3 Kalikow's auxiliary random walk

We denote for x, y ∈ Zd, P a Markov operator and δ < 1, the Green function of therandom walk killed at geometric rate 1− δ by

GPδ (x, y) := EP

x

[ ∞∑k=0

δk1Xk = y]and Gω

δ (x, y) := GPω

δ (x, y),

where P ω is the Markov operator associated with the random walk in the environmentω.

Then we introduce the so-called Kalikow environment associated with the point 0and the environment P1−ε[ · | I ], which is given for z ∈ Zd, δ < 1 and e ∈ ν by

pεδ(z, e) =E1−ε[G

ωδ (0, z)pω(z, e)|I]

E1−ε[Gωδ (0, z)|I]

.

103


The familly (pεδ(z, e))z∈Zd,e∈ν denes transition probabilities of a certain Markov chainon Zd. It is called Kalikow's auxiliary random walk and its rst appearence in a slightlydierent form goes back to [55].

This walk has proved to be useful because it links the annealed expectation of aGreen function of a random walk in random media to the Green function of a Markovchain. This result is summarized in the following proposition.

Proposition 3.1. For z ∈ Zd and δ < 1, we have

E1−ε

[Gωδ (0, z)|I

]= G

bpεδδ (0, z).

The proof of this result can be directly adapted from the proof of Proposition 1in [90]. We emphasize that in the case δ < 1, the uniform ellipticity condition is notneeded.

Using the former property we can link the Kalikow's auxiliary random walk to thespeed of our RWRE through the following proposition.

Proposition 3.2. For any 0 < ε < 1− pc(Zd), we have

limδ→1

∑z∈Zd G



= limδ→1

E[Xτδ ]

E[τδ]= v`(1− ε),

where dεδ(z) =∑e∈ν

pεδ(z, e)e.

Let Cεδ be the convex hull of all dεδ(z) for z ∈ Zd, then an immediate consequence of

the previous proposition is the following

Proposition 3.3. For ε > 0 we have that v`(1 − ε) is an accumulation point of Cεδ as

δ goes to 1.

The proofs of both propositions are contained in the proof of Proposition 2 in [90] andrely only on the existence of a limiting velocity, which is a consequence of Theorem 2.1.

In order to ease notations we will from time to time drop the dependence with respectto ε of the expectation E1−ε[ · ].

Let us now give a quick sketch of the proof of the continuity of the speed. A way ofunderstanding dεδ(z) is to decompose the expression of Kalikow's drift according to thepossible congurations at z

dεδ(z) =∑e∈ν

∑A⊂ν

E[1I1C(z) = AGω

δ (0, z)p(z, e)e]

E[1IGω

δ (0, z)] (3.1)

104

3. KALIKOW'S AUXILIARY RANDOM WALK

=∑

A⊂ν, A6=ν

E[1I1C(z) = AGω

δ (0, z)]

E[1IGω

δ (0, z)] dA

=∑

A⊂ν, A6=ν

P[C(z) = A]E[1IGω

δ (0, z)|C(z) = A]

E[1IGω

δ (0, z)] dA,

where dA =∑e∈A

pA(e)e is the drift under the conguration A.

Since P[C(z) = A] ∼ ε|A| for any A ∈ ν, if we want to nd the limit of dεδ(z) as ε goesto 0, it is natural to conjecture that the term corresponding to C(z) = ∅ is dominantin (3.1). For this, recalling the notations from (2.1), we may nd an upper bound on

E[1IGω

δ (0, z)|C(z) = A]

E[1IGω

δ (0, z)] =

E[1I(ωz,A)Gωz,A

δ (0, z)]

E[1IGω

δ (0, z)] , (3.2)

for z ∈ Zd, A ∈ 0, 1ν \ ν and δ < 1 which is uniform in z for δ close to 1, to be ableto apply Proposition 3.3 and show that |v`(1− ε)− d∅| = O(ε).

Let us show why the terms in (3.2) are upper bounded. It is easy to see that thedenominator is greater than γ1E[1I(ωz,1)Gωz,1

δ (0, z)], so we essentially need to showthat closing some edges adjacent to z cannot increase the quantity appearing in (3.2)by a huge amount. That is : for A ∈ 0, 1ν ,

E[1I(ωz,A)Gωz,A

δ (0, z)]≤ γ2E

[1I(ωz,1)Gωz,1

δ (0, z)]. (3.3)

In order to show that closing edges cannot have such a tremendous eect, let usrst remark that the Green function can be written as Gω

δ (0, z) = P ω0 [Tz < τδ]G

ωδ (z, z).

When we close some edges we might create a trap, for example a long corridorcanbe transformed into a dead-endand this eect can, in the quenched setting, increasearbitrarily Gω

δ (z, z), the number of returns to z .The rst step is to quantify this eect, we will essentially show in Section 4 that

Gωz,ν\A

δ (z, z) ≤ Gωz,1

δ (z, z)+Lz(ω) (see Proposition 4.2) where Lz(ω) is in some sense, tobe dened later, a localquantity around z (see Proposition 4.1 and Proposition 5.2).With this random variable we try to quantify how far from z the random walk has to goto nd a regularenvironment without traps where the eect of the modication aroundz is forgotten. In this upper bound, we may get rid of the term Gωz,1

δ (z, z) which is,once multiplied by 1IP ω

0 [Tz < τδ] ≤ 1I(ωz,1)P ωz,1

0 [Tz < τδ], of the same type as theterms on the right-hand side of (3.3).

The second step is to understand how the localquantity Lz is correlated with thehitting probability. The intuition here is that the hitting probability depends on the

105


environment as a whole but that a very local modication of the environment cannotchange tremendously the value of the hitting probability. On a more formal level thiscorresponds to (see Lemma 6.1) E[1IP ω

0 [Tz < τδ]Lz(ω)] ≤ γ3E[1IP ω0 [Tz < τδ]]

where γ3 is some moment of Lz, which is a sucient upper bound.Before turning to the proof, we emphasize that the aim of Section 4 and Section 5,

is mainly to introduce the so-called localquantities, which is done at the beginning ofSection 4, and prove some properties on these quantities, see Proposition 4.2, Proposi-tion 5.2 and Proposition 4.1. The corresponding proofs are essentially unrelated to therest of the paper and may be skipped in a rst reading, to concentrate on the actualproof of the continuity which is in Section 6.

4 Resistance estimates

In this section we shall introduce some elements of electrical networks theory (see [70])to estimate the variations on the diagonal of the Green function induced by a local mod-ication of the state of the edges around a vertex x. Our aim is to show that we can getecient upper bounds using only the local shape of the environment.

Let us denote the eective resistance between x ∈ Zd and a subgraph H ′ of a certainnite graph H by RH(x ↔ H ′). Denoting V (H ′) the vertices of H ′, it can be denedthrough Thomson's principle (see [70])

RH(x↔ H ′) = inf∑e∈H

r(e)θ2(e), θ(·) is a unit ow from x to V (H ′),

and this inmum is reached for the current ow from x to V (H ′). Under the environmentω, we will denote the resistance between x and y by Rω(x↔ y).

For a xed ω ∈ Ω, we add a cemetary point ∆ which is linked to any vertex xof K∞(ω) with a conductance such that at x the probability of going to ∆ is 1 − δand denote the associated weighted graph by ω(δ). We denote πω(δ)(x) the sum of theconductances of edges adjacent to x in ω(δ) and we dene Rω(δ)(x↔ ∆) to be the limitof Rω(δ)(x ↔ ω \ ωn) where ωn is any increasing exhaustion of subgraphs of ω. In thissetting we have,

πω(δ)(x) =πω(x)

δand rω(δ)([x,∆]) =

1

πω(δ)(x)

1

1− δ =1

πω(x)

δ

1− δ . (4.1)

We emphasize the fact that changing the state of an edge [x, y] changes the values ofrω(δ)([x,∆]) and rω(δ)([y,∆]), it can nevertheless be noted that Rayleigh's monotonicityprinciple (see [70]) is preserved, i.e. if we increase (resp. decrease) the resistance of oneedge the resistance of the graph the eective increases (resp. decreases).

There is no ambiguity to simplify the notations by setting Rω(x↔ ∆) := Rω(δ)(x↔∆) for x ∈ Zd and rω(e) := rω(δ)(e) for e any edge of ω(δ). It is classic (and can be foundas an exercice in chapter 2 of [70]) that

106

4. RESISTANCE ESTIMATES

Lemma 4.1. For any δ < 1, we have

Gωδ (x, x) = πω(δ)(x)Rω(x↔ ∆).

Hence to understand, in a rough sense, how closing edges might increase the numberof returns at z, we can concentrate on understanding the eect of closing edges onthe eective resistance. By Rayleigh's monotonicity principle, given a vertex x, theconguration in A = BE(x, r) which has the lowest resistance between any point and ∆is the one where all edges are open. Hence, for congurations B ∈ 0, 1A, we want toget an upper bound RωA,B(x↔ ∆) in terms of RωA,∅(x↔ ∆) and of localquantities.

Let us begin with a heuristic description of congurations which are likely to increasestrongly the number of returns when we close an edge. There are mainly two situationsthat can occur (see Figure 5.1)

1. the vertex x is in a long corridor, which is turned into a dead-endif we close onlyan edge, hence increasing the number of returns,

2. if closing an edge adjacent to x creates a new nite cluster K, the number ofreturns to x can be tremendously increased. Indeed because of the geometricalkilling parameter, when the particle gets stuck in K for a long time it may die(i.e. go to ∆), hence by closing the edge linking x to K, we can remove this escapepossibility and increase the number of returns to x.

~ℓ

K∞

deleted edge

infinite clusterx

K∞

∆

K

x

Fig. 5.1 Congurations where one deleted edge increases Gδ(x, x)

We want to nd properties of the environment which will quantify how strongly thenumber of returns will increase. In order to nd a quantity which controls the eect ofthe rst type of congurations we denote, for A = BE(x, r),

MA(ω) =

∞ if ∀y ∈ ∂A, y /∈ K∞(ωA,0),

maxy1,y2∈∂A∩K∞(ωA,0)

dωA,0(y1, y2) otherwise, (4.2)

107


which is the maximal distance between vertices of ∂A∩K∞(ωA,0) in the innite clusterof ωA,0. It is important to notice that the notation K∞(ωA,0) makes sense, since it isclassical that P-a.s. modifying the states of a nite set of edges does not create multipleinnite clusters.

This quantity will help us give upper bounds on the number of returns to x afterhaving closed some adjacent edges. Indeed even if the best escape way to innityisclosed, MA tells us in some sense how much more the particle has to struggle to getback onto this good escape route, even though some additional edges are closed.

In order to control the eect of the second type of bad congurations, we rst wantto nd out if we are likely to go to ∆ during an excursion into the part we called K. Forthis we introduce a way to measure the size of the biggest nite cluster of ωA,0 whichintersects ∂A,

TA(ω) =

0 if ∀y ∈ ∂A, y ∈ K∞(ωA,0),

maxy∈∂A, y/∈K∞(ωA,0)

|∂EKωA,0(y)| otherwise, (4.3)

which gives an indication on the time of an excursion into K, hence on the probabilityof going to ∆ during this excursion.

The idea now is to nd an alternate trap close to x, in which the walker is likely togo to ∆ before returning to x to replace the role of K. This alternate trap ensures thatthe number of return to x cannot be too big even in the case where all the accesses toparts such as K adjacent to x are closed. For this let us denote η ≥ 1 depending on dand ` such that

for all n ≥ 1, e2λ(η−1)n ≥ κ21(1 + |B(0, n)|), (4.4)

and H ′A(ω) the hyperplane y, y ·~≥ x·~+ηTA(ω). Any point of this hyperplane can beseen as an alternate trap since the particle is very unlikely to return to x when it reachesH ′A. Then a relevant quantity to control the eect of the second type of congurationsis the distance between x and this hyperplane, which quanties the diculty to reachan alternate strong trap.

In order to dene these quantities we need to know the innite clusterK∞, hence theyare not localquantities. Nevertheless we are able to dene random variables which arelocaland fulll the same functions. For A = BE(x, r), we denote L1

A(ω) the smallestpositive integer such that all y ∈ ∂A which are connected to ∂B(x, L1

A(ω)) in ωA,0,are connected to each other using only edges of BE(x, L1

A(ω)) ∩ ωA,0. We always haveL1A(ω) < ∞ by uniqueness of the innite cluster. Consequently there are two types of

vertices in ∂A, rst those which are not connected to ∂B(x, L1A(ω)) in ωA,0 (hence in a

nite cluster of ωA,0) and then those which are, the latter being all inter-connected inB(x, L1

A(ω)) ∩ ωA,0.Set HA(ω) to be the hyperplane y, y · ~≥ x · ~+ ηL1

A(ω) and nally let us deneLA(ω) the smallest integer such that

108


1. either ∂A is connected to HA(ω) using only edges of BE(x, LA(ω)) ∩ ωA,0,2. or ∂A is not connected to BE(x, LA(ω)), which can only happen if ∂A ∩K∞ = ∅,

and in order to make the notations lighter we use

Lz,k := LBE(z,k) and Lz = Lz,1. (4.5)

Using this denition for LA we get an upper bound for the quantities MA anddω(x,H ′A) on the event that x ∈ K∞ which is the only case we will need to consider.Now we can easily obtain, the proof is left to the reader, the following proposition

Proposition 4.1. For a ball A = BE(x, r), set Fx,n the σ-eld generated by ω(e), e ∈BE(x, n), we have the following

1. LA(ω) does not depend on the state of the edges in A,

2. LA(ω) is a stopping time with respect to (Fx,n)n≥0, in particular the event LA(ω) =k does not depend on the state of the edges of BE(x, k)c = E(Zd) \BE(x, k),

3. r ≤ LA(ω) <∞, P-a.s..

The second property is one of the two central properties for what we call a lo-calquantity. Recalling the notations (2.1) and (2.2), let us prove the following

Proposition 4.2. Set A = BE(x, r) with r ≥ 1, δ < 1 and ω ∈ Ω. Suppose thaty ∈ K∞(ω) and ∂A ∩K∞(ω) 6= ∅. We have

Rω(y ↔ ∆) ≤ RωA,1(y ↔ ∆) + C1LA(ω)C2e2λ(LA(ω)−x·~),

where C1 and C2 depend only on d and `.

Here the correcting term is essentially of the same order as the largest between

1. the resistance of paths linking the vertices of ∂A ∩K∞(ωA,0) inside B(x, LA),

2. the resistance of paths linking x to HA inside B(x, LA).

Démonstration. Let us introduce

A+ = B(x, r) ∪⋃

a∈∂A,a/∈K∞(ωA,0)

KωA,0(a) and A+,δ =⋃a∈A+

[a,∆],

moreover we set

A− = B(x, r − 1) ∪⋃

a∈∂A,a/∈K∞(ωA,0)

KωA,0(a) and A−,δ =⋃a∈A−[a,∆].

109


Let ωn be an exhaustion of ω and n0 such that B(x, LA(ω)) ∩ ω ⊂ ωn0 and y isconnected to ∂A in ωn0 . Set n ≥ n0, we denote θ(·) any unit ow from y to ω(δ) \ ωnusing only edges of ωn(δ). By Thomson's principle, we get

Rω(δ)(y ↔ ω(δ) \ ωn)−RωA,1(δ)(y ↔ ωA,1(δ) \ ωA,1n ) (4.6)

≤∑e∈ω(δ)

(rω(e)θ(e)2 − rωA,1(e)i0(e)2),

where i0(·) denotes the unit current ow from z to ωA,1(δ) \ ωA,1n . We want to applythe previous equation with a ow θ(·) which is close to the current ow i0(·). Since thelatter does not necessarily use only edges of ω we will need to redirect the part owingthrough A.

For a vertex a ∈ ∂A, we denote iA0 (a) =∑

e∈ν,[a,a+e]∈A i0([a, a + e]) the quantity ofcurrent entering A through a. Hence we can partition ∂A into

1. a1, . . . , ak the vertices of ∂A ∩K∞(ωA,0) such that iA0 (a) ≥ 0,

2. ak+1, . . . , al the vertices of ∂A ∩K∞(ωA,0) such that iA0 (a) < 0,

3. al+1, . . . , am the vertices of ∂A \K∞(ωA,0).

Moreover we denote

i+0 (∆) =∑e∈A+,δ

i0(e) and i−0 (∆) =∑e∈A−,δ

i0(e).

Let us rst assume y ∈ K∞(ωA,0), in particular y /∈ B(x, r − 1). The following factsare classical (see e.g. [70] chapter 2)

1. for any e ∈ E(Zd), we have |i0(e)| ≤ 1,

2. the intensity entering B(x, r − 1) is equal to the intensity leaving B(x, r − 1), i.e.∑i≤k

iA0 (ai) = i−0 (∆)−∑

j∈[k+1,l]

iA0 (aj).

Using the two previous remarks, we see it is possible to nd a collection ν(i, j) withi ∈ [1, k] and j ∈ [k + 1, l] ∪ ∆ such that

1. for all i, j, we have ν(i, j) ∈ [0, 1],

2. for all j ∈ [k + 1, l], it holds that∑

i≤k ν(i, j) = −iA0 (aj),

3. for all i ∈ [1, k], we have∑

j∈[k+1,l]∪∆ ν(i, j) = iA0 (ai),

4. it holds that∑

i≤k ν(i,∆) = i−0 (∆),

which should be seen as a way of matching the ow entering and leaving B(x, r − 1).Let us denote ~P(i, j) one of the directed paths between ai and aj in ωA,0∩BE(x, L1

A(ω)).Let ~Q be one of the directed paths from ∂A to HA(ω) in ωA,0 ∩ BE(x, LA(ω)) and let

110


us denote aj0 (with necessarily j0 ≤ l) its starting point and h1 its endpoint. The exis-tence of those paths is ensured by the denitions of L1

A, LA and HA and the fact that∂A ∩K∞ 6= ∅.

Finally let us notice that the values of the resistances rω([a,∆]) and rωA,1

([a,∆])might dier for a ∈ ∂A so that to get further simplications in (4.6), it is convenient toredirect the ow using these edges too. We introduce the unique ow (see Figure 5.2)dened by

θ0(~e) =

0 if e ∈ A+,δ,

0 if e ∈ E(A+),

i0(~e) + i+0 (∆) if ~e = [h1,∆],

i0(~e) +∑

i≤k,j∈[k+1,l] ν(i, j)1~e ∈ ~P(i, j)+∑

i≤k ν(i,∆)1~e ∈ ~P(i, j0)+ i+0 (∆)1~e ∈ ~Q+∑

i≤l i0([ai,∆])1~e ∈ ~P(i, j0) else.

y

B(x, LA)

A

∞h0

∆

Q

P(i, j)

P(k, j0)

Fig. 5.2 The ow θ0(·) in the case where y ∈ K∞(ωA,0)

In words, we could say that we have redirected parts of i0(·) in order to go aroundA and the ow going from A to ∆ is rst sent to aj0 , then to h1 and nally to ∆. Wehave the following properties

1. θ0(·) is a unit ow from y to ω(δ) \ ωn,2. |θ0(e)| ≤ 5 |∂A|2 for all e ∈ E(Zd),

3. θ0(·) coincides with i0(·) except on the edges of E(A+), A+,δ, Q, [h1,∆] and P(i, j)for i, j ≤ k + l ,

4. rω(·) coincides with rωA,1(·) except on the edges of E(A+) and A+,δ.

111


Hence recalling (4.6) we get

Rω(δ)(y ↔ ω(δ) \ ωn)−RωA,1(δ)(y ↔ ωA,1(δ) \ ωA,1n ) (4.7)

≤∑

e∈P(i,j)∪Q

rω(e)(θ0(e)2 − i0(e)2) + rω([h1,∆])(i+0 (∆) + i0([h1,∆]))2

−∑e∈A+,δ

rω(e)i0(e)2 − rω([h1,∆])i0([h1,∆])2

≤50ρd |∂A|6 LdAe2λ(LA−x·~) + rω([h1,∆])(i+0 (∆) + i0([h1,∆]))2

−∑e∈A+,δ

rω(e)i0(e)2 − rω([h1,∆])i0([h1,∆])2,

where we used that rω(e) ≤ e2λ(LA−x·~) for e ∈ P(i, j) ∪ Q and that there are at most(1 + |∂A|2)ρdL

dA ≤ 2ρd |∂A|2 LdA such edges in those paths. These properties being a

consequence of the fact that P(i, j) and Q are contained in BE(x, L1A(ω)).

Since |∂A| ≤ ρdrd ≤ ρdL

dA by the third property of Proposition 4.1, the rst term is

of the form announced in the proposition, the remaining issue is to control the remainingterms. First, we have by denition∑

e∈A+,δ

rω(e)i0(e)2 =∑a∈A+

rω([a,∆])i0([a,∆])2,

and since for a ∈ K∞(ω), we have using (4.1) and (2.5) that

κ1e−2λa·~ δ

1− δ ≥ rω([a,∆]) ≥ 1

κ1

e−2λa·~ δ

1− δ .

Furthermore, since for any a ∈ A+ we have a · ~≤ x · ~+ L1A and since h1 ∈ HA(ω)

we have h1 · ~≥ x · ~+ ηL1A ≥ a · ~+ (η− 1)L1

A so that the denition of η at (4.4) yields

1

κ1

e−2λa·~ ≥ 1

κ1

e−2λh1·~e2λ(η−1)L1A(ω) ≥ κ1(1 +

∣∣B(0, L1A)∣∣)e−2λh1·~.

Since A+ is contained in B(x, L1A(ω)), the two previous equations yield

rω([a,∆]) ≥ κ1(1 +∣∣A+

∣∣)e−2λh1·~ δ

1− δ ≥ (1 +∣∣A+

∣∣)rω([h1,∆]),

and hence ∑e∈A+,δ

rω(e)i0(e)2 + rω([h1,∆])i0([h1,∆])2

≥rω([h1,∆])(1 +∣∣A+

∣∣)(i0([h1,∆])2 +∑e∈A+

i0(e)2)

112


≥rω([h1,∆])(i+0 (∆) + i0([h1,∆]))2,

where we used Cauchy-Schwarz in the last inequality.Hence the remaining terms in (4.7) are non-positive and we have shown that

Rω(δ)(y ↔ ω(δ) \ ωn)−RωA,1(δ)(y ↔ ωA,1(δ) \ ωA,1n ) ≤ 50ρ7dL

7dA e

2λ(LA−x·~),

and letting n go to innity yields the result in the case where y ∈ K∞(ωA,0).Let us come back to the remaining case where y ∈ K∞(ω) \K∞(ωA,0). Keeping the

same notations, we see that obviously there exists j1 ≤ l such that aj1 ∈ K∞(ωA,0)

which is connected in ω to y using only vertices of A+, let us denote ~R path connectingaj1 and y in ω ∩ A+.

Introducing the unique ow (see Figure 5.3) dened by

θ′0(~e) =

1 if ~e ∈ ~R,0 if e ∈ A+,δ ∪ E(A+) \ R,i0(∆) + i0([h1,∆]) if ~e = [h1,∆]

i0(~e) +∑

j≤l iA0 (j)1~e ∈ ~P(j1, j)

+∑

i≤l i0([ai,∆])1~e ∈ ~P(i, j0)+i−0 (∆)1~e ∈ ~P(j1, j0)+ i+0 (∆)1~e ∈ ~Q else,

for which we can get the same properties as for θ0(·).

P(j1, j0)

R

Q h0∆

∞P(i, j)

y

A

B(x, LA)

Fig. 5.3 The ow θ′0(·) in the case where y ∈ K∞(ω) \K∞(ωA,0)

The computation of the energy of θ′0(·) is essentially similar to that of θ0(·) and weget

Rω(δ)(y ↔ ω(δ) \ ωn)−RωA,1(δ)(y ↔ ωA,1(δ) \ ωA,1n )

113


≤CL7dA e

2λ(LA−x·~) +∑e∈R

rω(e)

≤CL7dA e

2λ(LA−x·~),

since |R| ≤ |A+| ≤ ρdLdA and rω(e) ≤ e2λ(LA−x·~) for e ∈ R. The result follows.

We set for x, y ∈ Zd and Z ⊂ Zd,

Gδ,Z(x, y) = Eωx

[ TZ∑k=0

δk1Xk = y], (4.8)

and similarly we can dene Rω(x↔ Z∪∆) to be the limit of Rω(δ)(x↔ Z∪ω(δ)\ωn)where ωn is any increasing exhaustion of subgraphs of ω. We can get

Lemma 4.2. For any δ < 1, we have for x, z ∈ Zd,

Gωδ,z(x, x) = πω(δ)(x)Rω(x↔ z ∪∆).

In a way similar to the proof of Proposition 4.2, we get

Proposition 4.3. Set A = BE(x, r), B ∈ 0, 1A, δ < 1, z ∈ Zd and ω ∈ Ω. Supposethat y, z ∈ K∞(ω) and ∂A ∩K∞(ω) 6= ∅. We have

Rω(y ↔ z ∪∆) ≤ RωA,1(y ↔ z ∪∆) + C1LA(ω)C2e2λ(LA(ω)−x·~),

where C1 and C2 depend only on d and `.

We assume without loss of generality the constants are the same as Proposition 4.2.

Démonstration. This time let us denote i0(·) by the unit current ow from y to z ∪ω(δ) \ ωn.

The case where z ∈ K∞(ωA,0) can be treated using the same ows as in the proof ofProposition 4.2 and we will not give further details.

In order to treat the case where z /∈ K∞(ωA,0) and y ∈ K∞(ωA,0). We keep thenotations of the previous proof for the partition (ai)1≤i≤m of ∂A, i+0 (∆), i−0 (∆), A+ andA+,δ. We set

iz0 =∑e∈ν

i0([z + e, z]).

Similarly, we can nd a familly ν(i, j) with i ∈ [1, k] and j ∈ [k + 1, l] ∪ ∆ ∪ zsuch that

1. for all i, j, we have ν(i, j) ∈ [0, 1],

2. for all j ∈ [k + 1, l], it holds that∑

i≤k ν(i, j) = −iA0 (aj),

114

5. PERCOLATION ESTIMATE

3. we have∑

i≤k ν(i,∆) = i−0 (∆),

4. it holds that∑

i≤k ν(i, z) = iz0,

5. for all i ∈ [1, k] we have∑

j∈[k+1,l]∪∆∪z ν(i, j) = iA0 (ai).

We use again the same notations for P(i, j), Q, j0 and h1 and add an index j2 ≤ l

such that z is connected inside A+ to aj2 and ~S the corresponding directed path. Weset

θ0(~e) =

iz0(~e) if ~e ∈ ~S,0 if e ∈ A+,δ ∪ E(A+) \ S,i0(∆) + i0([h1,∆]) if ~e = [h1,∆]

i0(~e) + i+0 (∆)1~e ∈ ~Q+∑

i≤k,j∈[k+1,l] ν(i, j)1~e ∈ ~P(i, j)+∑

i≤k ν(i,∆)1~e ∈ ~P(i, j0)+∑

i≤k ν(i, z)1~e ∈ ~P(i, j2)+∑

i≤l i0([ai,∆])1~e ∈ ~P(i, j0) else,

which is similar to the ow considered in Proposition 4.2 except that the ow naturallysupposed to escape at z is, instead of entering A, redirected to aj2 and from theresent to z. Using this ow with Thomson's principle yields similar computations as inProposition 4.2 and thus we obtain a similar result.

The case where z /∈ K∞(ωA,0) and 0 /∈ K∞(ωA,0) can be easily adapted from theproof above and the second part of the proof of Proposition 4.2.

5 Percolation estimate

We want to give tail estimates on L1A and LA for some ball A = B(x, r). More

precisely we want to show for any C > 0, we have E1−ε[eCLA ] <∞ for ε small enough,

the exact statement can be found in Proposition 5.2. Let us recall the denitions of MA

and TA at (4.2) and (4.3). We see that all vertices of ∂A are either in nite clusters ofωA,0, which are included in B(x, r+ TA), or inter-connected in B(x, r+MA). Hence weget by the remarks above (4.5) that

L1A ≤ r + max(MA, TA). (5.1)

Recalling the denitions of LA and HA below (4.4), our overall strategy for provingthe existence of arbitrarily large exponential moments is the following : if LA is largethen there are two cases.

115


1. The random variable L1A is large. This means by (5.1) that eitherMA or TA is large.

The random variable TA cannot be large with high probability, since the distancein the percolation cluster cannot be much larger than the distance in Zd (seeLemma 5.2) and neither can MA since nite cluster are small in the supercriticalregime (see Lemma 5.3).

2. Otherwise the distance from x to HA in large in the percolation cluster is largeeven though it is not large in Zd. Once again this is unlikely, in fact for technicalreasons it appears to be easier to show that the distance to HA∩Tx is small, whereTx is some two-dimensional cone. For this we will need Lemma 5.5.

The following is fairly classical result about rst passage percolation with a minortwist due to the conditioning on the edges in A, we will outline the main idea of the proofwhile skiping a topological argument. To get a fully detailled proof of the topologicalargument, we refer the reader to the proof of Theorem 1.4 in [42].

Lemma 5.1. Set A = BE(x, r) and y, z ∈ Zd \B(x, r−1), there exists a non-increasingfunction α1 : [0, 1]→ [0, 1] such that for ε < ε1

P1−ε

[yωA,0↔ z, dωA,0(y, z) ≥ n+ 2dZd\B(x,r−1)(y, z)

]≤ 2α1(ε)n+dZd\B(x,r−1)

(y,z),

and

P1−ε

[yωA,0↔ z, dωA,0(y, z) ≥ n+ 2dZd(y, z) + 4dr

]≤ 2α1(ε)n+dZd (y,z),

where ε1 and α1(·) depend only on d and limε→0

α1(ε) = 0.

The main tool needed to prove Lemma 5.1 is a result of stochastic dominationfrom [63]. We recall that a familly Yu, u ∈ Zd of random variables is said to be k-dependent if for every a ∈ Zd, Ya is independent of Yu : ||u− a||1 ≥ k.Proposition 5.1. Let d, k be positive integers. There exists a non-decreasing functionα′ : [0, 1] → [0, 1] satisfying limτ→1 α

′(τ) = 1, such that the following holds : if Y =Yu, u ∈ Zd is a k-dependent familly of random variables satisfying

for all u ∈ Zd, P (Yu = 1) ≥ τ,

then PY (α′(τ)δ1 + (1− α′(τ))δ0)⊗Zd, where means stochasticaly dominated.

Two vertices u, v are ∗-neighbours if ||u− v||∞ = 1, this topology naturally inducesa notion of ∗-connected component on vertices.

Let us say that a vertex u ∈ Zd is ωA-wired if all edges [s, t] ∈ E(Zd) with ||u− s||∞ ≤1 and ||u− t||∞ ≤ 1 are open in ωA,1 (recall that A = BE(x, r)), otherwise it is calledωA-unwired.

We say that a vertex u ∈ Zd\B(x, r−1) is ωA-strongly-wired, if all y ∈ Zd\B(x, r−1)such that ||u− y||∞ ≤ 2 are ωA-wired, otherwise u is called ωA-weakly-wired. It is plain

116


that 1u is ωA-strongly-wired denes a γ1-dependent site percolation where γ1 dependsonly on d. We can thus use Proposition 5.1 with this familly of random variables sincewe have

for all u ∈ Zd, P1−ε[1u is ωA-strongly-wired = 1] ≥ (1− ε)γ1 ,

and that limε→0

(1− ε)γ1 = 1. This yields a function α′(·) which solely depends on d.

Démonstration. Let γ be one of the shortest paths in Zd \ B(x, r − 1) connecting y toz. For u ∈ Zd \B(x, r− 1), we dene V (u)(ωA) to be the ∗-connected component of theωA-unwired vertices of u and

V (ωA) =⋃u∈γ

V (u)(ωA).

Since y and z are connected in ωA,0, a topological argument (see Section 3 of [42]for details) proves there is an ωA,0-open path P from y to z using only vertices inγ ∪ (V (ωA,0) + −2,−1, 0, 1, 2d). On the event dωA,0(y, z) ≥ n+ 2dZd\B(x,r−1)(y, z), thispath P has m ≥ n + 2dZd\B(x,r−1)(y, z) + 1 vertices and all vertices which are not in γare ωA-weakly-wired thus there are at least m− dZd\B(x,r−1)(y, z)− 1 of them .

Since there are at most (2d)k paths of length k in Zd \ B(x, r − 1) we get, througha straightforward counting argument, that

P1−ε

[yωA,0↔ z, dωA,0(y, z) ≥ n+ dZd\B(x,r−1)(y, z)

]≤

∑m≥n+2dZd\B(x,r−1)

(y,z)+1

(2d)m(1− α′((1− ε)γ1))m−dZd\B(x,r−1)(y,z)−1

≤∑

m≥n+2dZd\B(x,r−1)(y,z)+1

((2d)3(1− α′((1− ε)γ1)))m−dZd\B(x,r−1)(y,z)−1,

where α′(·) is given by Proposition 5.1 and veries limε→0

1− α′((1− ε)γ1) = 0. Thus, the

rst part of the proposition is veried with α1(ε) := 1−α′((1−ε)γ1) and ε1 small enoughso that 1− α′((1− ε1)γ1) ≤ (2d)−3/2.

The second part is a consequence of

d(y, z) ≤ dZd\B(x,r−1)(y, z) ≤ d(y, z) + 2dr.

An easy consequence is the following tail estimate on MA (dened at (4.2)).

117


Lemma 5.2. Set A = BE(x, r), there exists a non-increasing function α1 : [0, 1]→ [0, 1]such that for ε < ε1

P1−ε[MA ≥ n+ 4dr] ≤ C3r2dα1(ε)n,

where C3, ε1 and α1(·) depend only on d and limε→0

α1(ε) = 0. The function α1(·) is the

same as in Lemma 5.1.

Démonstration. Since |∂A| ≤ ρdrd, we have

P1−ε[MA ≥ n+ 4dr]

≤(ρdrd)2 max

a,b∈∂AP1−ε

[aωA,0↔ b, dωA,0(a, b) ≥ n+ 4dr

]≤ γ1r

2dα1(ε)n,

where we used Lemma 5.1 since dZd\B(x,r−1)(a, b) ≤ 4dr for a, b ∈ ∂A.A set of n edges F disconnecting x from innity in Z, that is any innite simple path

starting from x uses an edge of F , is called a Peierls' contour of size n. Asymptoticson the number µn of Peierls' contours of size n have been intensively studied, see forexample [62], we will use the following bound proved in [89] and cited in [62],

µn ≤ 3n.

This enables us to prove the following tail estimate on TA (dened at (4.3)).


P1−ε[TA ≥ n] ≤ C4rdα2(ε)n,

where C4, ε2 and α2(·) depend only on d and limε→0

α2(ε) = 0.

Démonstration. First we notice that for n ≥ 1,

P1−ε[TA ≥ n] ≤ ρdrd maxa∈∂A

P1−ε

[a /∈ K∞(ωA,0),

∣∣∣∂EKωA,0(a)∣∣∣ ≥ n

].

For any a ∈ ∂A such that a /∈ K∞(ωA,0), we have that ∂EKωA,0(a) is a nite Peierls'

contour of size∣∣∣∂EKωA,0(a)

∣∣∣ surrounding a which has to be closed in ωA,0.

Because A is a ball at least half of the edges of ∂EKωA,0(a) have to be closed in ω aswell. Indeed, take [x, y] ∈ A∩∂EKωA,0(a) and denote x its endpoint in KωA,0(a), then bydenition of a Peierls' contour there is i ≥ 0 such that [x+i(x−y), x+(i+1)(x−y)] is in∂EK

ωA,0(a), let i0(x, y) be the smallest one. If [x+i0(x, y)(x−y), x+(i0(x, y)+1)(x−y)]were in A, since A is a ball all edges between x and x+ i0(x, y)(x− y) would too. Thiswould imply that all edges adjacent to y are in A which would contradict the fact thaty is connected to a ∈ ∂A in ωA,0. See Figure 5.4 for a two dimensionnal drawing.

118


∂EA

A

∂EKωA,0(a)

a

KωA,0(a)

[x, y]

[x + i0(x, y)(x− y), x + (i0(x, y) + 1)(x− y)]

Fig. 5.4 Half of the edges of ∂EKωA,0(a) have to be closed in ω

Hence

ψ :

A ∩ ∂EKωA,0(a) → ∂EK

ωA,0(a) \ A[x, y] 7→ [x+ i0(x, y)(x− y), x+ (i0(x, y) + 1)(x− y)],

is an injection so that at least half of the edges of ∂EKωA,0(a) are indeed closed in ω.There are at most

(mdm/2e

) ≤ γ12m ways of choosing those edges, thus we get for anya ∈ ∂A

P1−ε

[a /∈ K∞(ωA,0),

∣∣∣∂EKωA,0(a)∣∣∣ ≥ n

]≤∑m≥n

(m

dm/2e)µ(n)εm/2

≤ γ1

∑m≥n

6mεm/2 ≤ γ2εn/2,

for ε such that ε1/2 < 1/12.

A direct consequence of (5.1), Lemma 5.2 and Lemma 5.3 is the following tail esti-mate on L1

A, dened below (4.4)


P1−ε[L1A ≥ n+ C5r] ≤ C6r

2dα3(ε)n,

where C5, C6, ε3 and α3(·) depend only on d and limε→0

α3(ε) = 0.

119


Recalling the denition of HA above (4.5), let us introduce

L′A(ω) =

∞ if ∀y ∈ ∂A, y /∈ K∞(ωA,0)

dωA,0(∂A,HA(ω)) otherwise,(5.2)

it is plain that LA ≤ L′A.We need one more estimate before turning to the tail of L′A (and thus LA). Dene

the cone T = ae(1) + be(2), 0 ≤ b ≤ a/2 for a, b ∈ N. It is a standard percolationresult that pc(T) < 1 (see Section 11.5 of [46]) and well-known that the innite clusteris unique. We denote KT

∞(ω) the unique innite cluster of T induced by the percolationω, provided ε < 1− pc(T).

Lemma 5.5. There exists a non-increasing function α4 : [0, 1]→ [0, 1] so that for ε < ε4

P1−ε

[dT(0, KT

∞(ω)) ≥ 1 + n]≤ C7α4(ε)n,

where C7, α4(·) depend only on d and limε→0

α4(ε) = 0.

Démonstration. Choose ε < 1 − pc(T), so that KT∞(ω) is well dened almost surely.

We emphasize that the following reasoning is in essence two dimensional, so we areallowed to use duality arguments (see [46], Section 11.2). We recall that an edge of thedual lattice (i.e. of Z2 + (1/2, 1/2)) is called closed when it crosses a closed edge of theoriginal lattice.

If dT(0, KT∞(ω)) = n+ 1, then let x be a point for which this distance is reached, x is

one among at most n+2 possible points. Consider an edge e = [x, y] where dT(0, y) = n,let e′ denote the corresponding edge in the dual lattice. From each endpoint of this edgethere is a closed path in the dual lattice which has to cross out of T, so that the sum ofthe lengths of these two paths is at least n− 1. Thus there has to be a closed path P inthe dual lattice of length m ≥ b(n+1)/2c (including e′ in the largest of the previous twopaths) starting from one of the endpoints of e′ and crossing out of T (see Figure 5.5).

Thus since there are at most 4m paths of length m, we get for ε small enough

P1−ε

[dT(0, KT

∞(ω)) = 1 + n]≤ 2(n+ 2)

∑m≥n

2

4mεm ≤ 4(n+ 2)(4ε)n/2,

and the result follows from n+ 2 ≤ 2n+1 since

P1−ε

[dT(0, KT

∞(ω)) ≥ 1 + n]≤∑m≥n

4(n+ 2)(4ε)n/2 ≤ γ1(4ε1/2)n.

Now we turn to the study of the asymptotics of LA.

120


P closed in the dual lattice

endpoints of e

open edges

K∞(ω)

Fig. 5.5 The closed path in the dual lattice

Proposition 5.2. Set A = BE(x, r), there exists a non-increasing function α : [0, 1]→[0, 1] so that for ε < ε0

P1−ε[LA ≥ n+ C8r] ≤ C9r2dnα(ε)n,

where C8, C9, ε0 and α(·) depend only on d and ` and limε→0

α(ε) = 0.

Démonstration. Let us notice that two cases emerge. First let us consider that we areon the event ∂A ∩K∞ = ∅ in which case we have

LA(ω) = minn ≥ 0, ∂A is not connected to ∂B(x, n) ≤ r + TA(ω) ≤ L1A,

hence because of Lemma 5.4 we have for C8 > C5

P[∂A ∩K∞ = ∅, LA ≥ n+ C8r] ≤ C6r2dα3(ε)n. (5.3)

We are now interested in the case where ∂A ∩ K∞ 6= ∅. It is sucent to give anupper bound for L′A (dened at (5.2)) since LA ≤ L′A. Set ε < ε1∧ ε2∧ ε3∧ ε4, we noticeusing Lemma 5.4 that

P1−ε[∂A ∩K∞ 6= ∅, L′A ≥ n+ C8r] (5.4)

≤P1−ε[L1A ≥ n/(8ηd) + C5r]

+ P1−ε[∂A ∩K∞ 6= ∅, L1A ≤ n/(8ηd) + C5r, L

′A ≥ n+ C8r]

≤P1−ε[∂A ∩K∞ 6= ∅, L1A ≤ n/(8ηd) + C5r, L

′A ≥ n+ C8r]

121


+ C6r2dα1(ε)n/(8ηd).

We denote hxm the hyperplane y, y · ~≥ x · ~+m, we have

P1−ε[∂A ∩K∞ 6= ∅, L1A ≤ n/(8ηd) + C5r, L

′A ≥ n+ C8r] (5.5)

≤P1−ε

[∂A ∩K∞ 6= ∅, dωA,0(∂A, hxn/(8d)+ηC5r

) ≥ n+ C8r]

≤ |∂A|maxy∈∂A

P1−ε

[yωA,0↔ ∞, dωA,0(y, hxn/(8d)+γ1r

) ≥ n+ C8r].

Set y ∈ ∂A and let us denote γ2 a constant which will be chosen large enough. Usingthe uniqueness of the innite cluster we get

P1−ε

[dZd(y,K∞(ωA,0) ∩ hxn/(8d)+γ1r

∩ y + T)≥ n/2 + γ2r

](5.6)

≤P1−ε

[dy+T

(y,Ky+T

∞ (ωA,0) ∩ hxn/(8d)+γ1r

)≥ n/2 + γ2r

]≤P1−ε

[dy+T

(y,Ky+T

∞ (ω) ∩ hxn/(8d)+γ1r

)≥ n/2 + γ2r

],

where we have to suppose that γ2 ≥ 2 for the last inequality. Indeed then dy+T(y,Ky+T∞ (ω)) =

dy+T(y,Ky+T∞ (ωA,0)) on the event dy+T(y,Ky+T

∞ (ω)) ≥ γ2r since the distance to theinnite cluster is greater than the radius of A.

Moreover since e(1) · ~≥ 1/√d, we notice that

miny∈hxm∩T

dx+T(x, y) ≥ 2√dm.

Applying this for m = n/(8d) + γ1r, we get that

P1−ε

[dy+T

(y,Ky+T

∞ (ω) ∩ hyn/(8d)+γ1r

)≥ n/2 + γ2r

](5.7)

≤P1−ε

[dy+T

(y,Ky+T

∞ (ω))≥ n/2 + γ2r

],

where γ2 is large enough so that 2√d(n/(8d) + γ1r) ≤ n/2 + γ2r.

The equations (5.6) and (5.7) used with Lemma 5.5 yield that for γ3 large enoughand any y ∈ ∂A,

P1−ε

[dZd(y,K∞(ωA,0) ∩ hyn/(8d)+γ1r

∩ y + T)≥ n/2 + γ3r

]≤ γ4α4(ε)n/2.

If we use Lemma 5.1 and the previous inequality, for C8 large enough so that n +C8r ≥ 2(n/2 + γ3r) + 4dr,

P1−ε

[∂A ∩K∞ 6= ∅, dωA,0

(y, hyn/(8d)+γ1r

)≥ n+ C8r

](5.8)

122

6. CONTINUITY OF THE SPEED AT HIGH DENSITY

≤P1−ε

[dZd(y,K∞(ωA,0) ∩ hyn/(8d)+γ1r

∩ y + T)≥ n/2 + γ3r

]+

∑z∈∂BZd (y,dn/2+γ3re)∩y+T

P1−ε

[zωA,0↔ y, dωA,0(z, y) ≥ 2d(y, z) + 4dr

]≤γ4α4(ε)n/2 + γ5(n+ γ3r)α1(ε)n/2 ≤ γ6rnα5(ε)n,

where ε < ε5 depends only on d and ` for some α5(·) such that limε→0

α5(ε) = 0.

Adding up (5.4),(5.5) and (5.8) we get

P1−ε[∂A ∩K∞ 6= ∅, L′A ≥ n+ C8r] ≤ γ7nrd(α1(ε)n/(8ηd) + α5(ε)n)

≤ γ8nrdα(ε)n,

where α(ε) := α1(ε)1/(8ηd) +α5(ε). As we have limε→0

α(ε) = 0, this last equation and (5.3)

completes the proof of Proposition 5.2.

Essentially by replacing (Zd, E(Zd)) by (Zd, E(Zd \ [z, z + e])) and ω by ωz,e alongwith some minor modications we obtain

Proposition 5.3. Set A = BE(x, r), z ∈ Zd and e ∈ ν, there exists a non-increasingfunction α : [0, 1]→ [0, 1] so that for ε < ε0

P1−ε[LA(ωz,e) ≥ n+ C8r] ≤ C9rdnα(ε)n,

where C8, C9, ε0 and α(·) depend only on d and ` and limε→0

α(ε) = 0.

Here we assume without loss of generality that the constants are the same as inProposition 5.2.

6 Continuity of the speed at high density

We now have the necessary tools to study the central quantities which appearedin (3.2).

Proposition 6.1. For 0 < ε < ε5, A ∈ 0, 1ν \ ν and δ ≥ 1/2

E1−ε

[1I(ωz,A)Gωz,A

δ (0, z)]

E1−ε

[1IGω

δ (0, z)] < C,

where C and ε5 depend only on ` and d .

123


This section is devoted to the proof of this proposition. We have

E[1IGω

δ (0, z)]≥ E

[1C(z) = ∅1IGω

δ (0, z)]

= E[1C(z) = ∅1I(ωz,∅)Gωz,∅

δ (0, z)]

= P[C(z) = ∅]E[1I(ωz,∅)Gωz,∅

δ (0, z)].

For ε < 1/4 ≤ 1 − pc(d), we have P[C(z) = ∅] > γ1 > 0 for γ1 independent of ε, sothat

E[1IGω

δ (0, z)]≥ γ1E

[1I(ωz,∅)Gωz,∅

δ (0, z)]. (6.1)

Now we want a similar upper bound for the numerator of Proposition 6.1. Let A ∈0, 1ν \ ν, then by (2.5) and (4.1) we obtain

1

κ1

e2λz·~1

δ≤ πω

z,A(δ)(z) ≤ κ1e2λz·~1

δ. (6.2)

This equation combined with Lemma 4.1 yields

E[1I(ωz,A)Gωz,A

δ (0, z)]

(6.3)

≤κ1e2λz·~

δE[1I(ωz,A)P ωz,A

0 [Tz < τδ]Rωz,A(z ↔ ∆)

].

If z /∈ K∞(ωz,A) then P ωz,A

0 [Tz < τδ] = 0. Otherwise we can apply Proposition 4.2 toget

Rωz,A

δ (z ↔ ∆) ≤ Rωz,∅

δ (z ↔ ∆) + C1Lz(ω)C2e2λ(Lz(ω)−z·~), (6.4)

where we used notations from (4.5).Moreover we notice that P ωz,A

0 [Tz < τδ] ≤ P ωz,∅0 [Tz < τδ] and 1I(ωz,A) ≤ 1I(ωz,∅).

Then inserting (6.4) into (6.3), using Lemma 4.1 and (6.2) we get since δ ≥ 1/2

E[1IGω

δ (0, z)|C(z) = A]≤ κ2

1E[1I(ωz,∅)Gωz,∅

δ (0, z)]

(6.5)

+ 2C1κ1E[1I(ωz,∅)P ωz,∅

0 [Tz < τδ]Lz(ω)C2e2λLz(ω)].

Now we want to prove that the even though hitting probabilities depend on the wholeenvironment their correlation with localquantities are weak in some sense. Let us nowmake explicit the two properties which are crucial for what we call local quantitywhichare the second property of Proposition 4.1 and that arbitrarily large exponential mo-ments for ε small enough, like those obtained in Proposition 5.2. We obtain the followingdecorrelation lemma.

124


Lemma 6.1. Set δ ≥ 1/2, then

E[1I(ωz,∅)P ωz,∅

0 [Tz < τδ]Lz(ω)C2e2λLz(ω)]

≤C10E[1I(ωz,∅)P ωz,∅

0 [Tz < τδ]]E[Lz(ω)C11eC12Lz(ω)

],

where C10, C11 and C12 depend only on d and `.

Démonstration. First let us notice that the third property in Proposition 4.1 impliesthat Lz is nite. Set k ∈ N∗, recall that the event Lz = k depends only on edges inBE(z, k) by the second property of Proposition 4.1.

We have 1I(ωz,∅) ≤ 1∂B(z, k)↔∞. Assume rst that 0 /∈ B(z, k),


0 [Tz < τδ]Lz(ω)C2e2λLz(ω)|Lz = k]

(6.6)

=kC2e2λkE[1I(ωz,∅)P ωz,∅

0 [Tz < τδ] | Lz = k]

≤ρdkγ1e2λkE[1∂B(z, k)↔∞ max

x∈∂B(z,k)P ω

0 [Tx < τδ, Tx = T∂B(z,k)]],

indeed |∂B(z, k)| ≤ ρdkd, here we implicitely used that 0 /∈ B(z, k). Now the integrand

of the last term does not depend on the conguration of the edges in BE(z, k).We denote x0(ω) a vertex of ∂B(z, k) connected in ω to innity without using edges

of BE(z, k) and accordingly we introduce a ⇔ b the event that a is connected in ωto b using no edges of BE(z, k). Again we point out that the random variable x0(ω) ismeasurable with respect to ω(e), e /∈ BE(z, k).

Let us set x1(ω) the point for which the maximum in the last line of (6.6) is achieved,this random point also depends only on the set of congurations in E(Zd) \ BE(z, k),the same is true for P ω

0 [Tx0 < τδ, Tx0 = T∂B(z,k)].In case there are multiple choices the denition of the random variables x0(ω) or

x1(ω), we pick one of the choice according to some predetermined order on the verticesof Zd. In case x0(ω) or x1(ω) are not properly dened, i.e. when ∂B(z, k) is not connectedto innity, we set x0(ω) = z. With this denition we have x0 ⇔∞ = ∂B(z, k)↔∞.

The denition of x1 implies that

x1(ω)⇔ 0 if maxx∈∂B(z,k)

P ω0 [Tx < τδ, Tx = T∂B(z,k)] > 0.

Now let P0 be a path of k edges in Zd between z and x0 and P1 a path of k edges inZd between z and x1. As those paths are contained in BE(z, k), we get

E[1∂B(z, k)↔∞1x1 ⇔ 0P ω

0 [Tx1 < τδ, Tx1 = T∂B(z,k)]]

(6.7)

=E[1x0 ⇔∞1x1 ⇔ 0P ω

0 [Tx1 < τδ, Tx1 = T∂B(z,k)]|P0 ∪ P1 ∈ ω]

125


≤ 1

P[P0 ∪ P1 ∈ ω]E[1P0 ∪ P1 ∈ ω1x0 ⇔∞1x1 ⇔ 0P ω

0 [Tx1 < τδ]].

Then we see that since we have ε < 1/2

P[P0 ∪ P1 ∈ ω] = (1− ε)2k ≥ 1

4k. (6.8)

Moreover, on the event P0 ∈ ω, Markov's property yields

(δκ0)kP ω0 [Tx1 < τδ] ≤ P ω

0 [Tz < τδ].

Since δ ≥ 1/2,

E[1P0 ∪ P1 ∈ ω1x0 ⇔∞1x1 ⇔ 0P ω

0 [Tx1 < τδ]]

(6.9)

≤(2/κ0)kE[1P0 ∪ P1 ∈ ω1x0 ⇔ 01x1 ⇔∞P ω

0 [Tz < τδ]]

≤(2/κ0)kE[1IP ω

0 [Tz < τδ]],

since on 1P0 ∪ P1 ∈ ω1x0 ⇔ 01x1 ⇔∞ we have 0 ↔ x0 ↔ z ↔ x1 ↔ ∞ andwhich means that I occurs.

Adding up (6.6), (6.7), (6.8), (6.9), noticing that 1I ≤ 1I(ωz,∅) and P ω0 [Tz <

τδ] ≤ P ωz,∅0 [Tz < τδ], we get


0 [Tz < τδ]Lz(ω)C2e2λLz(ω) | Lz = k]

(6.10)

≤ρdkγ1(8e2λ/κ0)kE[1I(ωz,∅)P ωz,∅

0 [Tz < τδ]].

Let us come back to the case where 0 ∈ B(z, k). We can obtain the same result bysaying that P ωz,∅

0 [Tz < τδ] ≤ 1 in (6.6) and formally replacing P ω0 [Tx < τδ, Tx = T∂B(z,k)]

by 1 for any x ∈ ∂B(z, k) and x1 by 0 in the whole previous proof. The conclusion ofthis is that (6.10) holds in any case.

The result follows from an integration over all the events Lz = k for k ∈ N sinceby (6.10), we obtain


0 [Tz < τδ]Lz(ω)C2e2λLz(ω)]

≤E[ ∞∑k=1

P[Lz = k]E[1I(ωz,∅)P ωz,∅

0 [Tz < τδ]Lz(ω)C2e2λLz(ω) | Lz = k]]

≤ρdE[ ∞∑k=1

P[Lz = k]kγ1(8e2λ/κ0)kE[1I(ωz,∅)P ωz,∅

0 [Tz < τδ]]]

=ρdE[Lγ1z (8e2λ/κ0)Lz

]E[1I(ωz,∅)P ωz,∅

0 [Tz < τδ]].

126


We can apply Proposition 5.2 to get that for 0 < ε < ε6

E[Lz(ω)C11eC12Lz(ω)

]≤∑k≥0

kC11eC12kP[Lz ≥ k] < C13 <∞,

where ε6 is such that α0(ε6) < e−C12/2 and, as C13, depends only on d and `. Thenrecalling (6.5), using Lemma 6.1 with the previous equation we obtain

E[1I(ωz,A)Gωz,A

δ (0, z)]

≤κ21E[1I(ωz,∅)Gωz,∅

δ (0, z)]

+ 2C1C10C13κ1E[1I(ωz,∅)P ωz,∅

0 [Tz < τδ]]

≤γ2E[1I(ωz,∅)Gωz,∅

δ (0, z)].

Using the preceding equation with (6.1) concludes the proof of Lemma 6.1.We are now able to prove the following

Proposition 6.2. For any d ≥ 2, ε < 1− pc(d) and ` ∈ Rd we have

v`(1− ε) = d∅ +O(ε).

Démonstration. First notice that

P[C(z) = ∅] = 1 +O(ε) and P[C(z) 6= ∅] = O(ε).

using (3.1) and Lemma 6.1 we get for δ ≥ 1/2,

dεδ(z) = d∅E1−ε

[1I1C(z) = ∅Gω

δ (0, z)]

E1−ε

[1IGω

δ (0, z)] +O(ε), (6.11)

where the O(·) depends only on d and `. But using Lemma 6.1 again yields∣∣∣∣∣∣E1−ε

[1I1C(z) = ∅Gω

δ (0, z)]

E1−ε

[1IGω

δ (0, z)] − 1

∣∣∣∣∣∣ ≤ O(ε),

and thusdεδ(z)− d∅ = O(ε),

where the O(·) depends only on d and `. Recalling Proposition 3.3, we get

v`(1− ε) = d∅ +O(ε).

127


7 Derivative of the speed at high density

Next we want to obtain the derivative of the velocity with respect to the percolationparameter.

In this section we x z ∈ Zd. Using (3.1) with Proposition 6.1 we can get the rstorder of Kalikow's drift

dbωδ (z)− d∅ = ε

(∑e∈ν

E1−ε[1I(ωz,e)Gωz,e

δ (0, z)]

E1−ε[1I(ω)Gωδ (0, z)]

(de − d∅))

+O(ε2), (7.1)

where O(·) depends only on d and `. The remaining issue is the dependence of theexpectation with respect to ε.

7.1 Another perturbed environment of Kalikow

We can link the Green functions of two Markov operators P and P ′, since for n ≥ 0

GP ′

δ = GPδ +

n∑k=1

δk(GPδ (P ′ − P ))kGP

δ + δn+1(GPδ (P ′ − P ))n+1GP ′

δ . (7.2)

In our case we close one edge which changes the transition probabilities at two sites,so that the previous formula applied for n = 0,

Gωz,e

δ (0, z) =Gωz,∅

δ (0, z) + δGωz,∅

δ (0, z)∑e′∈ν

(pe(e′)− p∅(e′))Gωz,e

δ (z + e′, z) (7.3)

+ δGωz,∅

δ (0, z + e)∑e′∈ν

(p−e(e′)− p∅(e′))Gωz,e

δ (z + e+ e′, z),

where we used a notation from (2.3).Hence to compute the numerator of (7.1) using the expansion (7.3), we can look at

quantities such asE[1I(ωz,e)Gωz,∅

δ (0, z)Gωz,e

δ (z + e′, z)],

andE[1I(ωz,e)Gωz,∅

δ (0, z + e)Gωz,e

δ (z + e+ e′, z)],

for e, e′ ∈ ν.From now on we x e ∈ ν. In order to handle the rst type of terms (the proof is

similar for the second type of terms) we introduce the measure

dµz =1I1C(z) = eGωz,∅

δ (0, z)

E1−ε[1I1C(z) = eGωz,∅

δ (0, z)]dP1−ε,

128

7. DERIVATIVE OF THE SPEED AT HIGH DENSITY

and for e+ ∈ ν we introduce the Kalikow environment, corresponding to this measureon the environment and the point z + e+, dened by

pz,e,z+e+(y, e′) =Eµz [G

ωδ (z + e+, y)pω(y, y + e′)]

Eµz [Gωδ (z + e+, y)

=E1−ε[1IGωz,∅

δ (0, z)Gωδ (z + e+, y)pω(y, y + e′) | C(z) = e]

E1−ε[1IGωz,∅δ (0, z)Gω

δ (z + e+, y) | C(z) = e].

Once again Kalikow's property geometrically killed random walks does not use anyproperties on the measure on the environment, we have for any z ∈ Zd and e, e′ ∈ ν

E1−ε[1I1C(z) = eGz,∅δ (0, z)Gω

δ (z + e′, z)]

E[1I1C(z) = eGωz,∅δ (0, z)]

= Gpz,e,z+e′

δ (z + e′, z). (7.4)

Decomposing this according to the congurations at y and denoting Pz,e1−ε[ · ] =

P1−ε[ · | C(z) = e], we get

pz,e,z+e+(y, e′) =∑

A∈0,1νPz,e[C(y) = A] (7.5)

× Ez,e1−ε[1IGωz,∅

δ (0, z)Gωδ (z + e+, y) | C(y) = A]

Ez,e1−ε[1IGωz,∅

δ (0, z)Gωδ (z + e+, y)]

pA(y, e′),

where the conditional expectation is set to be 0 when it is not well-dened.In the next proposition we will use a+ = 0 ∨ a and from now on we will omit the

subscript in pz,e,z+e+ .

Proposition 7.1. For ε < ε7 and z, e, e+ ∈ Zd × ν2 and δ ≥ 1/2, we have for y ∈ Zd,e′ ∈ ν

|p(y, e′)− pz,e0 (y, e′)| ≤ (C14eC15((z−y)·~)+)ε,

where ε7, C14 and C15 depends on ` and d.

This proposition will be used to link the Green function of pz,e,z+e+ to the one ofpz,e0 . In view of (7.5) the previous proposition comes from

Proposition 7.2. For 0 < ε < ε8, y, z ∈ Zd, A ∈ 0, 1ν \ ν,

E1−ε[1I(ωz,e)Gωz,∅

δ (0, z)Gωz,e

δ (z + e+, y) | C(y) = A]

E1−ε[1I(ωz,e)Gωz,∅δ (0, z)Gωz,e

δ (z + e+, y) | C(y) = ∅] ≤ C16eC17((z−y)·~)+ ,

for ε8, C16, C17 depending only on ` and d for δ ≥ 1/2.

129


In order to prove Proposition 7.1, once we have noticed that we haveP[C(y) = ∅] ≥ γ1

and

Ez,e1−ε[1IGωz,∅

δ (0, z)Gωδ (z + e+, y)]

≥P[C(y) = ∅]E1−ε[1I(ωz,e)Gωz,∅

δ (0, z)Gωz,e

δ (z + e+, y) | C(y) = ∅],it suces to use the Proposition 7.2 with arguments close to the ones appearing in theproof of Proposition 6.2 to show that the dominant term in (7.5) is the one correspondingto C(y) = ∅.

Obviously Proposition 7.2 has strong similarities with Proposition 6.1, since the onlydierence is that the upper bound is weaker, which is simply due to technical reasons.Moreover, since the proof is rather technical and independent of the rest of the argument,we prefer to defer it to Section 8.

7.2 Expansion of Green functions

Once Proposition 7.1 is proved, we are able to approximate the Green functionsthrough the same type of arguments as given in [90]. Heuristically, we may say thatif environments are close then the Green functions should be close at least on shortdistance scales. Compared to [90], there is a twist due to the fact that we do not haveuniform ellipticity and that our control on the environment in Proposition 7.1 is onlyuniform in the direction of the drift. Moreover our limiting environmentas ε goes to 0is not translation invariant (nor uniformly elliptic). Hence we need some extra work toadapt the methods of [90].

Proposition 7.3. For any z ∈ Zd, e, e′ ∈ ν, e′′ ∈ ν ∪ 0 we get for δ ≥ 1/2∣∣∣Gpδ(z + e′ + e′′, z)−Gωz,e0

δ (z + e′ + e′′, z)∣∣∣ ≤ oε(1),

where oε(·) depends only on ` and d.

Démonstration. The proof will be divided in two main steps :

1. prove that there exists transition probabilities p that are uniformly close to thosecorresponding to the environment ωz,e0 on the whole lattice and which has a Greenfunction close to the one of the environment ω,

2. prove the same statement as in Proposition 7.3 but for the environment p. This isinspired from the proof of Lemma 3 in [90].

For the rst step, we will show that the random walk is unlikely to visit often z andgo far away in the direction opposite to the drift, i.e. we want to show that for anyε1 > 0

Gpδ(z + e′, z)−Gp

δ,x∈Zd, x·~<z·~−A(z + e′, z) < ε1 for A large and ε small,

130


where we used a notation of (4.8). This inequality comes from the fact that except at zand z + e the local drift under p can be set to be uniformly positive in the direction ìn any half-space x ∈ Zd, x · ` > −A for ε small.

In a rst time, we show that the escape probability from z and z + e′ is lowerbounded in the environment p. For this we can easily adapt a classical super-martingaleargument, see Lemma 1.1 in [97], to get that for any η > 0 there exists f(η) > 0such that for any random walk on Zd dened by a Markov operator P (x, y) such that(∑

y∼x P (x, y)(y − x)) · ~ > η, for x such that x · ~≥ 0, we have

P0[Xn · ~≥ 0, n > 0] > f(η). (7.6)

Now by Proposition 7.1, it is possible to x a percolation parameter 1 − ε where εis chosen small enough so that

The drift d(p)(x) =∑

e∈ν p(x, x + e)e is such that dp(x) · ~ > d∅ · ~/2 for x such

that x · ~ ≥ (z + 2de(1)) · ~ (this way we avoid the transitions probabilities at thevertices z and z + e which are special).

The transition probabilities on the shortest paths from z and z + e′ to z + 2de(1)

(with length inferior to some γ1) are greater than κ0/2.Hence we can get a lower bound for the escape probability under p :

miny∈z,z+e′

P py [T+z,z+e′ =∞] (7.7)

≥ miny∈z,z+e′

P py [T+z,z+e′ > Tz+2de(1) ]P

p

z+2de(1)[(Xn − (z + 2de(1))) · ~≥ 0, n > 0]

≥f(d∅ · ~/2)(κ0

2

)γ1= γ2.

Now we need to show that the walk is unlikely to go far to the left. Considerany random walk on Zd given by a transition operator P (x, y) such that dP (x) :=∑

y∼x P (x, y)(y − x) · ~ > (d∅ · ~)/2 = γ3. We know that

MPn = Xn −X0 −

n−1∑i=1

dP (Xi),

is a martingale with jumps bounded by 2. Hence since dP (x) ≥ γ3, we can use Azuma'sinequality, see [4], to get

P0[Tx∈Zd, x·~<−A <∞] ≤∑n≥0

P0

[MP

n · ~ < −A− γ3n]

≤∑n≥0

exp(−(A+ γ3n)2

8n

)≤ γ4 exp

(−γ3A

4

).

131


Set ε′ > 0 small enough. Taking A = A(ε′) large enough, depending also on d and `,we can make the right-hand side lower than ε1. Now let us choose ε small enough so thatfor any y ∈ x ∈ Zd, x · ~≥ z · ~−A−1 and x 6= z, z+e′ we have dω(y) · ~ > (d∅ · ~)/2.Then the previous imply

maxy∈z,z+e′

P py [Tx∈Zd, x·~<z·~−A−1 < T+

z,z+e′] (7.8)

≤ maxy∼z or y∼z+e′

P py [Tx∈Zd, x·~<z·~−A−1 < T+

z,z+e′] ≤ ε′,

since the event in the middle depends only on the transitions probabilities at the verticesof x ∈ Zd, x · ` ≥ z · ~− A− 1 and x 6= z, z + e′.

Moreover by (7.7) we have

Gpδ(z + e′, z + e′) = P p

z+e′ [T+z+e′ < τδ]

−1 ≤ 1

γ2

and Gpδ(z, z) ≤ 1

γ2

, (7.9)

which decomposing with respect to the number of excursions to z and z + e′ and us-ing (7.8) yields

P pz+e′ [Tx∈Zd, x·~<z·~−A−1 <∞] <

2ε′

γ2

. (7.10)

For ε small enough to verify the previous conditions we have using (7.9) and (7.10)

Gpδ(z + e′, z)−Gp

δ,x∈Zd, x·~<z·~−A−1(z + e′, z) (7.11)

≤P pz+e′ [Tx∈Zd, x·~<z·~−A−1 <∞] max

y∈x∈Zd, x·~<z·~−A−1Gpδ(y, z)

≤2ε′

γ2

Gpδ(z, z) =

2ε′

γ22

.

So that introducing ω(y, e′) so that

p(y, e′) = p(y, e′) for y such that (y − z) · ~≥ −A(ε′)

p(y, e′) = pωz,e0 (y, e′) for y such that (y − z) · ~ < −A(ε′)

A consequence of (7.11) is that∣∣∣Gpδ(z + e′, z)−Gp

δ(z + e′, z)∣∣∣ ≤ γ5ε

′, (7.12)

where, by Proposition 7.1, ω (depending on ε′) is such that

maxe′∈ν,y∈Zd

∣∣∣p(y, e′)− pωz,e0 (y, e′)∣∣∣ ≤ C14e

C15A(ε′)ε ≤ ε′, (7.13)

for ε small enough. This completes step (1).

132


We can start step (2) of the proof. Since our control on the environment is uniformand our remaining task is to use methods similar to those of [90] to prove that thereexists a o(·) depending only on d and ` such that∣∣∣Gωz,e0

δ (z + e′ + e′′, z)−Gpδ(z + e′ + e′′, z)

∣∣∣ = o(1), (7.14)

which in view of (7.12) is enough to prove Proposition 7.3.Let us dene M the operator of multiplication by (πω

z,e0 )1/2 given for f : Zd → R, by

M(f)(y) = (πωz,e0 (y))1/2f(y).

We consider a transition operator P s,δ of a random walk on Zd ∪ ∆ given by

P s,δ(x, x+ e(i)) = P s,δ(x+ e(i), x)

= δ(πωz,e0 (x))1/2pω

z,e0 (x, x+ e(i))(πω

z,e0 (x+ e(i)))−1/2

= δ(πωz,e0 (x+ e(i)))1/2pω

z,e0 (x+ e(i), x)(πω

z,e0 (x))−1/2,

for any i = 1, . . . , 2d and for x /∈ z, z + e,∆

P s,δ(x,∆) = (1− δ) + δ∑e(i)∈ν

(p∅(e(i))p∅(−e(i)))1/2,

P s,δ(z,∆) = (1− δ) + δ∑

e(i)∈ν\e

(pe(e(i))pe(−e(i)))1/2,

P s,δ(z + e,∆) = (1− δ) + δ∑

e(i)∈ν\−e

(p−e(e(i))p−e(−e(i)))1/2,

and P s,δ(∆,∆) = 1.

We have for x, y 6= ∆,

Gωz,e0δ (x, y) = ((I − δP ωz,e0 )−1)(x, y) = (M−1(I − P s,δ)−1M)(x, y)

= (M−1Gs,δM)(x, y),

where Gs,δ is the Green function of P s,δ. In a similar way, we have for x, y 6= ∆

Gpδ(x, y) = (M−1Gs,δM)(x, y),

where because of (7.13) the associated operator P s,δ = M−1P pM veries

P s,δ(x, x+ e) = P s,δ(x, x+ e) + εξε(x, e)

and P s,δ(x,∆) = P s,δ(x,∆) + εξε(x,∆),

133


where ξε(·, ·) are uniformly bounded.Now, we expand the Green function, using (7.2) to obtain that for any n

Gpδ(x, x

′)−Gωz,e0δ (x, x′) =

n∑i=1

(δε)kSk(x, x′) + (δε)n+1Rn(x, x′),

where

Sn(x, x′) =∑

x1,...,xn

∑e1,...,en

Gωz,e0δ (x, x1)ξε(x1, e1)G

ωz,e0δ (x1 + e1, x2) · · ·

× ξε(xn, en)Gωz,e0δ (xn + en, x

′),

andRn(x, x′) =

∑x∗∈Zd

Sn(x, x∗)∑e∗∈ν

ξε(x∗, e∗)Gp

δ(x∗ + e∗, x′).

Consider the transformation

Sn(x, x′) =(πωz,e0 (x′)

πωz,e0 (x)

)1/2 ∑x1,...,xne1,...,en

Gs,δ(x, x1)ξε(x1, e1)( πω

z,e0 (x1)

πωz,e0 (x1 + e1)

)1/2

×Gs,δ(x1 + e1, x2) · · ·( πω

z,e0 (xn)

πωz,e0 (xn + en)

)1/2

Gs,δ(xn + en, x′),

and

Rn(x, x′) =(πωz,e0 (x′)

πωz,e0 (x)

)1/2 ∑x1,...,xne1,...,en

Gs,δ(x, x1)ξε(x1, e1)( πω

z,e0 (x1)

πωz,e0 (x1 + e1)

)1/2

×Gs,δ(x1 + e1, x2) · · ·( πω

z,e0 (xn)

πωz,e0 (xn + en)

)1/2

Gs,δ(xn + en, x′).

We have for any x ∈ Zd and δ ≥ 1/2,

∣∣1− P s,δ(x,∆)∣∣ ≥ δ

2minek∈ν

d∑j=1

(√pek(e(j))−

√pek(−e(j))

)2

= γ6.

Moreover for any x ∈ Zd and ei ∈ ν we get by (2.5) that

πωz,e0 (x)

πωz,e0 (x+ ei)

≤ κ21e

2λ,

and for x, x′ ∈ Zd we obtain∑x1,...,xne1,...,en

Gs,δ(x, x1)Gs,δ(x1 + e1, x2) · · ·Gs,δ(xn + en, x′)

134


≤(∑

x1

Gs,δ(x, x1)(2d))

maxx∗∈Zd

∑x2,...,xne2,...,en

Gs,δ(x∗, x2) . . . Gs,δ(xn + en, x′)

≤ 2d

1−maxx P s,δ(x,∆)maxx∗∈Zd

∑x2,...,xne2,...,en

Gs,δ(x∗, x2) . . . Gs,δ(xn + en, x′)

≤ · · · ≤( 2d

1− γ6

)n,

where we used an easy recursion to obtain the last inequality. Finally we get

Sn(x, x′) =(πωz,e0 (x′)

πωz,e0 (x)

)1/2(κ2

1e2λ(

supy,e|ξε(y, e)|

) 2d

1− γ6

)n+1

≤(πωz,e0 (x′)

πωz,e0 (x)

)1/2

γn+17 ,

for some positive constant γ7, depending only on d and `. We can get a similar estimatefor the remaining term Rn(x, x′) considering that 1− P s,δ(x,∆) ∼ 1− P s,δ(x,∆). Thisimplies that for ε < γ−1

7 /2 small enough, the series∑∞

k=0(εδ)kSk(x, x′) is convergent

and upper bounded by a constant independent of δ and that

Gpδ(x, x

′)−Gωz,e0δ (x, x′) =

∞∑k=1

(δε)kSk(x, x′)

=(πωz,e0 (x′)

πωz,e0 (x)

)1/2

O(ε),

where O(·) depends only on d and `.Applying this last result for all cases x = z + e′ + e′′ and x′ = z yields (7.14) and

thus the result.

7.3 First order expansion of the asymptotic speed

We have now all the necessary tools to compute the asymptotic speed. ApplyingProposition 7.3 (which relies on Proposition 7.1), we get

Gpz,e,z+e′

δ (z + e′, z) = Gωz,e0δ (z + e′, z) + o(1),

where the o(·) depends only on d and `. Hence putting the previous equation togetherwith (7.4), we obtain

E1−ε[1I1C(z) = eGωz,∅

δ (0, z)Gωδ (z + e′, z)] (7.15)

=(1 + o(1))E[1I1C(z) = eGωz,∅

δ (0, z)]Gω0,e

0δ (e′, 0),

135


where the o(·) depends only on d and `.Applying the same methods for pz+e,−e,z+e+e′ yields

E[1I(ωz,e)Gωz,∅

δ (0, z + e)Gωz,e

δ (z + e+ e′, z)] (7.16)

= (1 + o(1))E[1I(ωz,e)Gωz,∅

δ (0, z + e)]Gω0,e

0δ (e+ e′, 0),

where the o(·) depends only on d and `.Let us denote

φ(e) =∑e′∈ν

(pe(e′)− p∅(e′))Gω0,e0 (e′, 0),

andψ(e) =

∑e′∈ν

(p−e(e′)− p∅(e′))Gω0,e0 (e+ e′, 0).

Hence imputing the estimates (7.15) and (7.16) into the expression of (7.1) modiedusing (7.3), we get

dbωδ (z)− d∅ (7.17)

=ε(1 + o(1))

E1−ε[1IGωδ (0, z)]

[E1−ε[1I(ωz,e)Gωz,∅

δ (0, z)](∑e∈ν

(1 + δφ(e))(de − d∅))

+ E1−ε[1I(ωz,e)Gωz,∅

δ (0, z + e)](∑e∈ν

δψ(e)(de − d∅))]

+O(ε2),

where the o(·) and O(·) depend only on d and `.We are not able to derive uniform estimates for dbω

δ (z), nevertheless we are still ableto estimate the asymptotic speed.

Lemma 7.1. We have for δ ≥ 1/2, z ∈ Zd and e ∈ ν,∣∣∣∣∣E[1I(ωz,e)Gωz,∅

δ (0, z)]

E[1IGωδ (0, z)]

− 1

∣∣∣∣∣ ≤ O(ε),

where the O(·) depends only on d and `.

Démonstration. Recalling that P[C(z) = ∅] = 1 +O(ε), we get

E[1I(ωz,e)Gωz,∅

δ (0, z)]

E[1IGωδ (0, z)]

≤ 1

P[C(z) = ∅]E[1I(ωz,e)Gωz,∅

δ (0, z)]

E[1I(ωz,∅)Gωz,∅δ (0, z)]

≤ 1 +O(ε).

In order to prove the other bound we will use methods very similar to those usedbefore and we will not give a fully-detailed proof. In words the idea is that we cancondition the environment to be open on any nite ball without changing, up to terms

136


of order ε, the value of the expectation of Green functions. Hence, up to a O(ε), wemay suppose that BE(z, 2) contains only open edges and with this conditionning thetwo terms appearing in the quotient of the lemma are the same. With more formalism,we have

E[1I(ωz,e)Gωz,∅

δ (0, z)] =∑

A∈0,1BE(z,2)

A 6=∅

P[e ∈ BE(z, 2), e ∈ ω = A] (7.18)

× E[1I(ωz,e)Gωz,∅

δ (0, z) | e ∈ BE(z, 2), e ∈ ω = A],

it can be shown that for any A ⊂ BE(z, 2), A 6= ∅

E[1I(ωz,e)Gωz,∅

δ (0, z) | e ∈ BE(z, 2), e ∈ ω = A]

E[1I(ωz,e)Gωz,∅δ (0, z)]

< γ1, (7.19)

where γ1 depends only on d and `. The method is the same as before

1. We apply Lemma 4.1 to decompose the Green function into a hitting probabilityand a resistance.

2. With Lemma 4.2 we decompose the resistance appearing in (1) into the sameresistance in ωB

E(z,2),1 and a local perturbation.

3. By upper bounding the integrand by terms which do not depend on the state ofthe edges of BE(z, 2) and using the same trick as in (6.7) we prove that for ε smallenough

E[1I(ωz,e(z,2),A)P ωz,∅

(z,2),A

0 [Tz < τδ]Rω(z,2),1

[z ↔ ∆]]e2λz·~

E[1I(ωz,e)Gωz,∅δ (0, z)]

< γ2.

We can use arguments similar to the ones in the proof of Lemma 6.1 (essentiallyrepeating the steps (6.6), (6.7), (6.8) and (6.9)) to prove that

E[1I(ωz,e(z,2),A)P ωz,∅

(z,2),A

0 [Tz < τδ]LC2z,2e

2λLz,2

]E[1I(ωz,e)Gωz,∅

δ (0, z)]

=E[1I(ω(z,2),A)P ωz,∅

(z,2),A

0 [Tz < τδ]LC2z,2e

2λLz,2 | C(z) = e]

E[1IGωz,∅δ (0, z) | C(z) = e]

< γ3,

since Lz,2 has arbitrarily large exponential moments under the measure P[ · |C(z) = e], for ε small enough by Proposition 5.3. For the rst equality in the pre-vious equation, we implicitely used that Lz,2 does not depend on the congurationat z which is true by the rst property of Proposition 4.1.

137


This reasoning yields (7.19) with γ1 = γ2+C1γ3. Now, the equations (7.19) and (7.18)imply that

E[1I(ωz,e)Gωz,∅

δ (0, z)] = O(ε)E[1I(ωz,e)Gωz,∅

δ (0, z)]

+ E[1I(ωz,e)Gωz,∅

δ (0, z), 1e ∈ ω, e ∈ BE(z, 2) = BE(z, 2)],

and since on 1e ∈ ω, e ∈ BE(z, 2) = BE(z, 2) we have 1IGωδ (0, z) = 1I(ωz,e)Gωz,∅

δ (0, z),it follows that

E[1I(ωz,e)Gωz,∅

δ (0, z)]

=(1 +O(ε))E[1IGωδ (0, z), 1e ∈ ω, e ∈ BE(z, 2) = BE(z, 2)]

≤(1 +O(ε))E[1IGωδ (0, z)],

which yields the other bound.

Recalling Proposition 3.1, we can use (7.17) and Lemma 7.1 to get∑z∈Zd G



− d∅

=ε(1 + o(1))

∑z∈Zd

∑e∈ν E[1IGω

δ (0, z)](1 + δφ(e))(de − d∅)∑z∈Zd E[1IGω

δ (0, z)]

+ ε(1 + o(1))

∑z∈Zd

∑e∈ν E[1IGω

δ (0, z + e)]δψ(e)(de − d∅)∑z∈Zd E[1IGω

δ (0, z)]

=ε∑e∈ν

(1 + δ(φ(e) + ψ(e)))(de − d∅) + o(ε),

since∑

z E[1IGωδ (0, z)] =

∑z E[1IGω

δ (0, z + e)] = P[I]/(1− δ).In order to simplify the previous expression we prove

Lemma 7.2. We have∑e′∈ν

(pe(e′)− p∅(e′))Gω0,e0 (e′, 0) +

∑e′∈ν

(p−e(e′)− p∅(e′))Gω0,e0 (e+ e′, 0)

= (p∅(e)− p∅(−e))(Gω0,e0 (0, 0)−Gω0,e

0 (e, 0))− p∅(e).

Démonstration. Recalling the notations (2.3), we get

pe(e′)− p∅(e′) =

c(e′)c(e)π∅πe

if e 6= e′,

− c(e′)π∅

if e = e′.

138

8. ESTIMATE ON KALIKOW'S ENVIRONMENT

Hence we get that,∑e′ 6=e

c(e′)c(e)

π∅πeGω0,e

0 (e′, 0) =c(e)

π∅(Gω0,e

0 (0, 0)− 1), (7.20)

and ∑e′ 6=e

c(e′)c(−e)π∅π−e

Gω0,e0 (e+ e′, 0) =

c(−e)π∅

Gω0,e0 (e, 0). (7.21)

Finally using c(e)

π∅= p∅(e) and the previous equations, the computations are straight-

forward.

Recalling that p∅(e)−p∅(−e) = d∅ ·e and 1−p∅(e) = πe/π∅, we see that the previouslemma means that

α(e) = φ(e) + ψ(e) =πe

π∅+ (d∅ · e)(Gω0,e

0 (0, 0)−Gω0,e0 (e, 0)),

and so Proposition 3.2 yields by letting δ go to 1

v`(1− ε) = d∅ + ε∑e∈ν

α(e)(de − d∅) + o(ε). (7.22)

We still may simplify slightly the expression of the speed we obtained using thefollowing

∑e∈ν

πede =2d∑i=1

∑e 6=e(i)

c(e)e = (2d− 1)∑e∈ν

c(e)e = (2d− 1)π∅d∅ =∑e∈ν

πed∅,

Inserting this last equation into (7.22) yields

v`(1− ε) = d∅ + ε∑e∈ν

(d∅ · e)(Gω0,e0 (0, 0)−Gω0,e

0 (e, 0))(de − d∅) + o(ε),

which proves Theorem 2.2.

8 Estimate on Kalikow's environment

The section is devoted to the proof of Proposition 7.2 so that we still assume to havea xed z ∈ Zd and e, e+ ∈ ν. Before entering into the details let us present the mainsteps of the proof of the previous proposition which are rather similar to the ones in theproof of Proposition 6.1.

1. The Green functions behave essentially as a hitting probability multiplied by aresistance (normalized by the invariant measure). See (8.19).

139


2. Then we use the estimates on resistances of Proposition 4.2. This procedure willessentially give an upper bound of the numerator in Proposition 7.2 as a nitesum of terms which are linked to the denominator but with a local correlationaround y due to the presence of random variables similar to the random variableLzwhich appeared in the proof of Proposition 6.1. See (8.27) and (8.28).

3. We nish the proof by decorrelation lemmas similar to Lemma 6.1 to handle thesecond type of terms we just described. See sub-section 8.3

Compared to Proposition 6.1 there is an extra diculty added by the fact that weneed to handle two Green functions instead of only one (in some sense we will even havethree), hence we will apply Proposition 4.2 recursively, this is done in Proposition 8.2.

We point out that in addition, we cannot prove a decorrelation lemma involvingone of the hitting probabilities coming from the Green functions appearing in Proposi-tion 7.2. Hence we need to transform one of them into an expression which we will beable to decorrelate from a local modication of the environment and this will changeslightly the outline of the proof given above. The aim of the next sub-section is to takecare of this problem.

8.1 The perturbed hitting probabilites

We want to understand the eect of the change of conguration around y on the hit-

ting probabilities Pωz,ey,Az+e+ [Ty < τδ] and P

ωz,∅y,A0 [Tz ≤ τδ]. The former term can be estimated

since if we denote B∗(y, k) =t ∈ B(y, k), t

BE(y,k)\[z,z+e]↔ yand

pωz (y, k) =

maxu∈∂B∗(y,k)

P ωz+e+

[Tu = T∂B∗(y,k) < τδ] if z + e+ /∈ B∗(y, k),

1 otherwise.(8.1)

then for any k ≥ 1 such that 0 /∈ B∗(y, k), we have

Pωz,ey,Az+e+ [Ty < τδ] ≤ ρdk

dpωz,ey,Az (y, k), (8.2)

the special notation B∗(y, k) is useful because in the conguration ωz,e the walker canonly reach y by entering the ball B(y, k) through B∗(y, k).

As we announced previously, the second hitting probability is more dicult to treat.Let us introduce the following notations

pω1 (y, k) =

maxu∈∂B(y,k)

P ω0 [Tu = T∂B(y,k) < τδ] if 0 /∈ B(y, k),

1 otherwise,(8.3)

140


pω2 (y, k) =

maxu∈∂B(y,k)

P ωu [Tz < τδ ∧ T+

∂B(y,k)] if z /∈ B(y, k),

1 otherwise.(8.4)

To make notations lighter we also set

Rω∗ (z) = e2λz·~Rω[z ↔ ∆] and Rω

∗ (y) = e2λy·~Rω[y ↔ ∆], (8.5)

and moreover we introduce

Rω

∗ (y, k) =

maxu∈∂B(y,k)

Rω∗ [u↔ z ∪∆] if z /∈ B(y, k),

1 otherwise,(8.6)

wherefor any u ∈ Zd, Rω

∗ [u↔ z ∪∆] = e2λz·~Rω[u↔ z ∪∆]. (8.7)

then we have the following proposition.

Proposition 8.1. Take any conguration ω and set y, z ∈ Zd and B = B(y, r) withr ≥ 1 and δ ≥ 1/2. If 0, z /∈ B and P ω

0 [Tz < τδ] > 2P ω0 [Tz < T∂B ∧ τδ], then we have

P ω0 [Tz < τδ] ≤ C19r

2dpω1 (y, k)pω2 (y, k)Rω

∗ (y, k).

If 0 ∈ B, z /∈ B and P ω0 [Tz < τδ] > 2P ω

0 [Tz < T∂B ∧ τδ], thenP ω

0 [Tz < τδ] ≤ C19r2dpω2 (y, k)R

ω

∗ (y, k).

Finally if 0 /∈ B and z ∈ B,P ω

0 [Tz < τδ] ≤ C19r2dpω1 (y, k).

Thanks to this lemma we can say that P ω0 [Tz < τδ] is either not inuenced much

by a local modication around y (in the case where naturally the walk will not visity when it goes from 0 to z), or upper bounded by a product of at most three randomvariables. Two of them behave as hitting probabilities which are well suited for ourfuture decorrelation purposes, the third random variable is essentially a resistance forwhich we have estimates as well.

Démonstration. We will only consider the case 0, z /∈ B, the other being similar butsimpler. Our hypothesis implies

P ω0 [Tz < τδ] ≤ 2P ω

0 [T∂B ≤ Tz < τδ],

we can get an upper bound on the right-hand term by Markov's property

P ω0 [TB ≤ Tz < τδ] =

∑u∈∂B

P ω0 [Tu = T∂B < τδ]P

ωu [Tz < τδ] (8.8)

141


≤ |∂B|maxu∈∂B

P ω0 [Tu = T∂B < τδ] max

u∈∂BP ωu [Tz < τδ].

Denoting z1 → · · · → zn the event that the n rst vertices of ∂B ∪ z ∪∆ visited are,in order, z1, z2, . . . , zn, we can write for u ∈ ∂B

P ωu [Tz < τδ] = Eω

u

[∑n

∑z1,...,zn∈∂B

1z1 → · · · → zn → z]

(8.9)

=∑n

∑z1,...,zn∈∂B

Eωu [1z1 → · · · → zn]P ω

zn [Tz < T+∂B ∧ τδ]

≤ maxv∈∂B

P ωv [Tz < τδ ∧ T+

∂B]Eωu

[∑n

∑z1,...,zn∈∂B

1z1 → · · · → zn]

= maxv∈∂B

P ωv [Tz < τδ ∧ T+

∂B]Gωδ,z(u, ∂B),

where

Gωδ,z(u, ∂B) = Eω

u

[τδ∧Tz∑n=0

1Xn ∈ ∂B]≤ |∂B|max

v∈∂BGωδ,z(v, v). (8.10)

Since by Lemma 4.2, (2.5) and (4.1) we have for δ ≥ 1/2 and any v ∈ ∂B

Gωδ,z(v, v) = πω(δ)(v)Rω(v ↔ z ∪ δ) ≤ γ1 max

u∈∂BRω∗ [u↔ z ∪∆]. (8.11)

Since |∂B| ≤ ρdrd adding up (8.9), (8.10) and (8.11) we get

maxu∈∂B

P ωu [Tz < τδ] ≤ γ2r

d maxu∈∂B

Rω∗ [u↔ z ∪∆] max

u∈∂BP ωu [Tz < τδ ∧ T+

∂B].

Using the previous equation with (8.8) concludes the proof of the Proposition.

Recalling the notations from (2.1), (2.2) and (8.7) , let us introduce

Rω∗ (y, k) =

minu∈∂B(y,k)

Rω(y,k),1∗ [u↔ z ∪∆] if z /∈ B(y, k),

1 otherwise,(8.12)

For the future decorrelation part we need to rewrite Rω(y,r),1

∗ (y, r) in terms of Rω∗ (y, r

′)and local quantities. This is done in the following lemma.

Lemma 8.1. For any B = BE(y, r) and r′ ≥ r. Suppose that z ∈ K∞(ω) and ∂B ∩K∞(ω) 6= ∅, we have

Rω(y,r),1

∗ (y, r) ≤ e4λr′Rω∗ (y, r

′) + C20LC21

y,r′eC22Ly,r′ .

142


Démonstration. Let us denote v ∈ ∂B(y, r) such that

maxu∈∂B(y,k)

Rω(y,r),1∗ (u↔ z ∪∆) = Rω(y,r),1(v ↔ z ∪∆)e2λv·~, (8.13)

applying Proposition 4.3 we get for any r′ ≥ r

Rω(y,r),1∗ (v ↔ z ∪∆) ≤ Rω(y,r′),1(v ↔ z ∪∆) + C1L

C2

y,r′e2λ(−y·~+Ly,r′ ) (8.14)

≤ Rω(y,r′),1(v ↔ z ∪∆) + C1LC2

y,r′e2λ(−v·~+2Ly,r′ ),

where we used that y · ~ ≥ v · ~− r and that Ly,r′ ≥ r′ ≥ r by the third property ofProposition 4.1.

For any u ∈ ∂B(y, r′), let us denote i0(·) the unit current from u to z ∪ ∆ inω(y,r′),1 and ~Q one of the shortest directed path from v to u included in B(y, r′). Wecan use Thompson's principle applied to the unit ow from v to z ∪ ∆ given byθ(e) = i0(e) + (1e ∈ ~Q − 1−e ∈ ~Q) to get

Rω(y,r′),1(v ↔ z ∪∆) ≤ R

ωz,∅(y,r′),1

(u↔ z ∪∆) + 8r′e2λ(−y·~+r′), (8.15)

where we skipped a simple exhaustion argument.Hence adding up (8.14) and (8.15), we get

Rω(y,r),1∗ (v ↔ z ∪∆) ≤ min

u∈∂B(y,r′)Rω(y,r′),1∗ (v ↔ z ∪∆)

+ γ1(Ly,r′)γ2γ

Ly,r′3 e−2λy·~,

since Ly,r′ ≥ r′ ≥ r.We get multiplying the left side by e2λv·~ and the right one by e2λr′e2λy·~ (which is

greater than e2λv·~) that

Rω(y,r),1∗ (v ↔ z ∪∆)e2λv·~ ≤ e4λr′Rω

∗ (y, r′) + γ4(Ly,r′)

γ5eγ6Ly,r′

where we used that maxu∈∂B(y,r′) e2λu·~ ≤ e2λr′e2λy·~. So by (8.13) we obtain the lemma.

8.2 Quenched estimates on perturbed Green functions

The aim of this subsection is to complete the rst two steps of the sketch of proofat the beginning of Section 8. Let us introduce

R∗(z) = Rωz,∅y,A∗ (z) and R∗(y) = R

ωz,ey,A∗ (y), (8.16)

143


and this way we reduce our problem of studying Green functions to studying resistances,indeed using Lemma 4.1 and (6.2) we get for δ ≥ 1/2,

1

κ1

Gωz,∅y,Aδ (z, z) ≤ R∗(z) ≤ 2κ1G

ωz,∅y,Aδ (z, z), (8.17)

and1

κ1

Gωz,ey,Aδ (y, y) ≤ R∗(y) ≤ 2κ1G

ωz,ey,Aδ (y, y). (8.18)

Moreover we can now easily obtain the rst step of our proof since

Gωz,∅y,Aδ (0, z)G

ωz,ey,Aδ (z + e+, y) ≤ 4κ2

1Pωz,∅y,A0 [Tz ≤ τδ]P

ωz,ey,Az+e+ [Ty ≤ τδ]R∗(z)R∗(y). (8.19)

As mentionned before, we will apply recursively the resistance estimates of Proposi-tion 4.2, for this we introduce

l(0)y = 1, l(1)

y = Ly,1, l(2)y = L

y,l(1)y

and l(3)y = L

y,l(2)y,

L(i)y (ω) = l

(i)y (ωz,∅) ∨ l(i)y (ωz,e) and B(i)

y = BE(y, L(i)y ). Moreover we set

Zy,k = C23kC24eC25ke2λ((z−y)·~)+ and Z(i)

y = Zy,L

(i)y,

where and C23 = 1∨C1 ∨C19 ∨C20, C24 = C2 ∨C21 ∨ 2d and C25 = 4λ∨C22. Moreoverset, for i = 0, . . . , 3

R(i)∗ (y) = R

ωz,e

B(i)y ,1

∗ (y) and R(i)∗ (z) = R

ωz,∅

B(i)y ,1

∗ (z). (8.20)

Also recalling (8.12), we set

R(i)∗ = Rωz,∅

∗ (y, L(i)y ) and R

(i)

∗ = Rωz,∅

∗ (y, L(i)y ).

Finally we denote for i = 1, 2, 3 and j = 0, 1, 2

p(j)i = pω

z,∅

i (y, L(j)y ) and p(j)

z = pωz,e

z (y, L(j)y ).

In particular, since we assumed Z(i)y large enough, we can get from Proposition 8.1

that for any z ∈ Zd and i ∈ 0, 1, 2,

Pωz,ey,A0 [Tz < τδ] ≤ Z(i)

y p(i)1 p

(i)2 R

(i)

∗ + 2Pωz,ey,A0 [Tz < T

∂B(i)y∧ τδ], (8.21)

and also from Lemma 8.1 we obtain that for any y, z ∈ K∞(ωz,ey,A), we have for any i ≤ j

R(i)

∗ ≤ Z(j)y R(j)

∗ + Z(j+1)y . (8.22)

144


Moreover for y, z ∈ K∞(ωz,ey,A), the random variables Z(i)y are large enough for us to

apply Proposition 4.2 so that we have for i ∈ 0, 1, 2R∗(z) ≤ R(i)

∗ (z) + Z(i+1)y and R∗(y) ≤ R(i)

∗ (y) + Z(i+1)y . (8.23)

The equations (8.21) and (8.22) yield that for any y, z ∈ K∞(ωz,ey,A)

Pωz,ey,A0 [Tz < τδ]R∗(z)R∗(y) (8.24)

≤(p

(0)1 p

(0)2 (Z(0)

y R(0) + Z(1)y ) + 2P

ωz,ey,A0 [Tz < T

∂B(0)y∧ τδ]

)R∗(z)R∗(y).

The idea is now to use recursively (8.23), (8.21) and (8.22) to control all the previousterms. We can obtain the following proposition

Proposition 8.2. For any ω such that y, z ∈ K∞(ωz,ey,A),

R(0)R∗(z)R∗(y) ≤C26

[R(0)∗ R

(0)∗ (z)R(0)

∗ (y)

+ (Z(1)y )2(R(1)

∗ R(1)∗ (z) +R(1)

∗ R(1)∗ (z) +R(1)

∗ (z)R(1)∗ (y))

+ (Z(2)y )4(R(2)

∗ +R(2)∗ (z) +R(2)

∗ (y)) + (Z(3)y )4

],

andR∗(z)R∗(y) ≤ C27

[R(1)∗ (z)R(1)

∗ (y) + Z(2)y (R(2)

∗ (z) +R(2)∗ (y)) + (Z(3)

y )2].

This is an interesting upper bound since the resistances are only multiplied withindependent local quantities, e.g. Z(2)

y is independent of R(2)∗ (y), since Z(2)

y depends onlyon the stopping timeL(2)

y , i.e. only on the edges of BE(y, L(2)y ) by the second property

of Proposition 4.1.

Démonstration. Let us prove the rst upper bound, we use (8.23) to get

R(0)∗ R∗(z)R∗(y) ≤ R(0)

∗ (R(0)∗ (z) + Z(1)

y )(R(0)∗ (y) + Z(1)

y )

≤R(0)∗ R

(0)∗ (z)R(0)

∗ (y) + Z(1)y (R(0)

∗ R(0)∗ (z) +R(0)

∗ R(0)∗ (y)) + (Z(1)

y )2R(0)∗ .

The rst term of the right-hand side is of the form announced in the proposition.We need to simplify the remaining terms, we will continue the expansion for R(0)

∗ R(0)∗ (z)

(the method is similar for R(0)∗ R

(0)∗ (y)), using (8.23) and (8.22)

R(0)∗ R

(0)∗ (z)

≤(Z(1)y R(1)

∗ + Z(2)y )(R(1)

∗ (z) + Z(2)y )

≤Z(1)y R(1)

∗ R(1)∗ (z) + Z(2)

y (Z(1)y R(1)

∗ +R(1)∗ (z)) + (Z(2)

y )2

≤Z(1)y R(1)

∗ R(1)∗ (z) + (Z(2)

y )2(Z(2)y R(2)

∗ +R(2)∗ (z) + 2Z(3)

y ) + (Z(2)y )2

145


≤Z(1)y R(1)

∗ R(1)∗ (z) + (Z(2)

y )3(R(2)∗ +R(2)

∗ (z)) + 3(Z(3)y )3,

where we used that for any i ≤ j we have 1 ≤ Z(i)y ≤ Z

(j)y . All terms here are of the

same type as in the proposition.The expansion for the term (Z

(1)y )2R(0)

∗ is handled by applying (8.22) for i = 0 andj = 1. Once again our upper bound is correct.

The second upper bound is similar and simpler since it uses only (8.23), so we skipthe details.

Recalling the notations (8.3), (8.4) and (8.1) we have for j ∈ z, 1, 2

for y ∈ Zd and k1 < k2, pωj (y, k1) ≤ ρdkd−12 pωj (y, k2), (8.25)

so that for k1 < k2 ∈ 0, 1, 2,

p(k1)j ≤ ρdZ

(k2)y p

(k2)j . (8.26)

Finally let us take notice of the trivial inequality P ω0 [Tz < T

∂B(0)y∧ τδ] ≤ P ω

0 [Tz <

T∂B

(i)y∧ τδ] which used with (8.25) and Proposition 8.2 can expand (8.24). We can apply

Proposition 8.2 which can be applied since y, z ∈ K∞(ωz,ey,A) if the integrand is positive.Then, imputing this expanded equation into (8.19)and using (8.2) we can show that it

is possible to give an upper bound on Gωz,∅y,Aδ (0, z)G

ωz,ey,Aδ (z, y) with a nite sum of terms

of the form1I(ωy,A)(Z(i)

y )C28p(i)z p

(i)1 p

(i)2 R

(i)∗ R

(i)∗ (z)R(i)

∗ (y), (8.27)

and

1I(ωy,A)(Z(i)y )C28p(i)

z Pωz,∅y,A0 [Tz < T

∂B(i)y∧ τδ]R(i)

∗ (y)R(i)∗ (z), (8.28)

for i ∈ 0, 1, 2, 3 and also similar terms where R(i)∗ , R

(i)∗ (z) or R(i)

∗ (y) are possiblyreplaced by 1.

This is what we aimed for at the beginning of the proof of Proposition 7.2 : thecorrelation term Z

(i)y is associated only with terms with which it is independent. Now

we are only left with the third step of our proof that is the decorrelation part. Indeed,

since we need to give an upper bound on E[Gωz,∅y,Aδ (0, z)G

ωz,ey,Aδ (z, y)

], we shall look for an

upper bound on the expectations of (8.27) and (8.28), which is the subject of the nextsub-section.

8.3 Decorrelation part

Let us prove the rst decorrelation lemma.

146


Lemma 8.2. We have for i ∈ 0, 1, 2, 3 and δ ≥ 1/2

E[1I(ωy,A)(Z(i)

y )C28p(i)z p

(i)1 p

(i)2 R

(i)∗ R

(i)∗ (z)R(i)

∗ (y)]

≤C29E[(L(i)

y )C30eC31L(i)y

]E[1I(ωy,∅)Gωz,∅

y,∅δ (0, z)G

ωz,ey,∅δ (z + e+, y)

]eC32((y−z)·~)+ ,

where C29, C30, C31 and C32 depend only on d and `.

The same lemma holds, with dierent constants, if we replace R(i)∗ , R

(i)∗ (z) or R(i)

∗ (y)by 1. Indeed it can be seen using Rayleigh's monotonicity principle that for δ > 1/2,these three quantities are lower bounded by

1 ∧Rω0(0) ∧mink∈N

minu∈∂B(0,k), z /∈B(0,k)

Rω0∗ (u↔ z ∪∆) ≥ γ1,

where γ1 can be chosen independent of y, i and z. By Lemma 4.2,

Rω0∗ (u↔ z ∪∆) ≥ γ1G

ω0

δ,z(u, u) = γ1Pω0u [T+

u < Tz ∧ τδ]−1

≥ P ω0u [τδ > 2, X1 = y,X2 = u]−1 ≥ γ1κ

20/4,

where y is some neighbour of u which is not z.

Démonstration. First let us notice that if the integrand is positive then 1I(ωy,A)p(i)1

is positive and necessarily

0↔ ∂B(i)y , 0

ωB(i)y ,0

↔ ∞which we denote 0⇔ y ⇔∞, we

recall L(i)y <∞ by Proposition 4.1.

Let us condition on the event L(i)y = k for k <∞. First suppose that 0 /∈ B(y, k),

z /∈ B(y, k) and z + e+ /∈ B∗(y, k), where we used a notation appearing above (8.1).Recalling the notations (8.1), (8.3) and (8.4), we may denote x0 ∈ ∂B∗(y, k) and x1, x2 ∈∂B(y, k) such that

pωz (y, k) = P ωz,e

z+e+[Tx0 = T∂B∗(y,k) < τδ],

pω1 (y, k) = P ωz,∅

0 [Tx1 = TB(y,k) < τδ],

pω2 (y, k) = P ωz,∅

x2[Tz < τδ ∧ T+

B(y,k)],

where x0 is connected to y in BE(y, k)\[z, z+e] and we denote P0 one of the correspond-ing shortest such paths (hence of length ≤ k + 2). Moreover let us set x3 connecting∂B(y, k) to innity without edges of B(y, k). Thus

E[1I(ωy,A)(Z(i)

y )C28p(i)z p

(i)1 p

(i)2 R

(i)∗ R

(i)∗ (y)R(i)

∗ (z) | L(i)y = k

]≤γ1k

γ2eγ3keγ4((z−y)·~)+E[10⇔ y ⇔∞p(i)

z p(i)1 p

(i)2 R

(i)∗ R

(i)∗ (y)R(i)

∗ (z) | L(i)y = k

],

147


where the integrand of the right-hand side depends only on the edges of E(Zd)\BE(y, k),so that the conditionning is unnecessary.

Let us denote P1, P2 and P3 one of the shortest paths from respectively x1, x2 andx3 to y and P = P0 ∪ P1 ∪ P2 ∪ P3 ∪ y + e, e ∈ ν. Hence we need to control

E[10⇔ y ⇔∞p(i)

z p(i)1 p

(i)2 R

(i)∗ R

(i)∗ (z)R(i)

∗ (y) | L(i)y = k

]≤E[10⇔ y ⇔∞P ωz,e

z+e+[Tx0 = T∂B∗(y,k) < τδ]P

ωz,∅

0 [Tx1 = T∂B(y,k) < τδ]

× P ωz,∅

x2[Tz < τδ ∧ T+

∂B(y,k)]Rω(y,k),1

∗ [x2 ↔ z ∪∆]R(0)∗ (z)R(0)

∗ (y) | P ∈ ω]

≤24k+4d+2E[10⇔ y ⇔∞1P ∈ ωP ωz,∅

y,∅0 [Tx1 = T∂B(y,k) < τδ]R

(0)∗ (z)R(0)

∗ (y)

× P ωz,∅y,∅

x2 [Tz < τδ ∧ T+∂B(y,k)]P

ωz,ey,∅

z+e+ [Tx0 = T∂B∗(y,k) < τδ]Rωz,∅y,∅∗ [x2 ↔ z ∪∆]

],

where we used that

1. P[P ∈ ω] ≥ 2−(4k+4d+2),

2. inequalities such as P ωz,e

z+e+[Tx0 = T∂B∗(y,k) < τδ] = P

ωz,ey,∅

z+e+ [Tx0 = T∂B∗(y,k) < τδ],

3. Rayleigh's monotonicity principle yields, for example, that

R(i)∗ (y) ≤ R(0)

∗ (y) and Rωz,∅

(y,k),1∗ [x2 ↔ z ∪∆] ≤ R

ωz,∅y,∅∗ [x2 ↔ z ∪∆].

Using (4.3) and x2 ∈ B(y, k), we get

Pωz,∅y,∅

x2 [Tz < τδ ∧ T+∂B(y,k)]R

ωz,∅y,∅∗ [x2 ↔ z ∪∆]

≤γ5Pωz,∅y,∅

x2 [Tz < τδ ∧ T+x2

](Pωz,∅y,∅

x2 [Tz ∧ τδ < T+x2

])−1

=γ5Pωz,∅y,∅

x2 [Tz < τδ ∧ T+x2| Tz ∧ τδ < T+

x2]

=γ5Pωz,∅y,∅

x2 [Tz < τδ],

moreover on ω such that 1P ∈ ω,

Pωz,∅y,∅

x1 [Tx2 < τδ] ≥ (δκ0)2k,

and putting these last two equations together we get

Pωz,∅y,∅

0 [Tx1 = T∂B(y,k) < τδ]Pωz,∅y,∅


ωz,∅y,∅∗ [x2 ↔ z ∪∆]

≤γ5(δκ0)−2kPωz,∅y,∅

0 [Tx1 < τδ]Pωz,∅y,∅

x1 [Tx2 < τδ]Pωz,∅y,∅

x2 [Tz < τδ]

148


≤γ5(δκ0)−2kPωz,∅y,∅

0 [Tz < δ].

In a similar way, we get by Markov's property that

Pωz,ey,∅

z+e+ [Tx0 = T∂B∗(y,k) < τδ] ≤ (δκ0)−(k+2)Pωz,ey,∅

z+e+ [Ty < τδ].

Finally10⇔ y ⇔∞1P ∈ ω ≤ 1I.

Hence for ω such that P ∈ ω we have

10⇔ y ⇔∞1P ∈ ωP ωz,∅y,∅

0 [Tx0 = T∂B(y,k) < τδ]Pωz,ey,∅

z+e+ [Tx0 = T∂B∗(y,k) < τδ]

× P ωz,∅y,∅


ωz,∅y,∅∗ [x2 ↔ z ∪∆]R(0)

∗ (z)R(0)∗ (y)

≤γ6(2/κ0)3k1IP ωz,∅y,∅

0 [Tz < τδ]Pωz,ey,∅

z+e+ [Ty < δ]R(0)∗ (z)R(0)

∗ (y)

≤γ7eγ8k1IGωz,∅

y,∅δ (0, z)G

ωz,ey,∅δ (z + e+, y),

where we used Rayleigh's monotonicity principle to say that R(0)∗ (y) ≤ Rωz,e

y,∅ [y ↔∆], (8.17) and (8.18).

The result follows by integrating over all possible values of L(i)y , since we have just

proved that

E[1I(ωy,A)(Z(i)

y )C28p(i)z p

(i)1 p

(i)2 R

(i)∗ R

(i)∗ (z)R(i)

∗ (y) | L(i)y = k

]≤γ9k

γ10eγ11kE[1I(ωy,∅)Gωz,∅

y,∅δ (0, z)G


]eγ12((y−z)·~)+ .

For the remaining cases, we proceed as follows

1. if 0 ∈ B(y, k), then we formally replace P ω0 [Tx = T∂B(z,k) < τδ] by 1 for any

x ∈ ∂B(z, k) and x1 by 0,

2. if z + e+ /∈ B∗(y, k), then we formally replace P ωz+e+ [Tx = T∂B∗(z,k) < τδ] by 1 for

any x ∈ ∂B∗(z, k) and x0 by z + e+,

3. if z ∈ B(y, k), then we formally replace P ωx [Tz = T+

∂B(z,k) ∧ τδ] by 1 for any x ∈∂B(z, k), R

ωz,∅y,∅∗ [x2 ↔ z ∪∆] by 1 and x2 by z,

and the previous proof carries over with minor modications.

We need another decorrelation lemma, which is essentially similar to the previousone but simpler to prove.

149


Lemma 8.3. We have for i ∈ 0, 1, 2, 3 and δ ≥ 1/2,

E[1I(ωy,A)(Z(i)

y )C28p(i)z P

ωz,∅y,A0 [Tz < T


∗ (y)R(i)∗ (z)

]≤C33E

[(L(i)

y )C34eC35L(i)y

]E[1I(ωy,∅)Gωz,∅

y,∅δ (0, z)G


]eC36((y−z)·~)+ ,

where the constants depend only on d and `.

Démonstration. Once again we condition on L(i)y = k for k < ∞ and suppose that

0 /∈ B(y, k) and z /∈ B(y, k), the other cases can be handled in the same way as before.We see that

1I(ωy,A) ≤ 10⇔ y ⇔∞,and we denote x0, x1 ∈ ∂B(y, k) such that

p(i)z = P ωz,e

z [Tx0 = T∂B(y,k) < τδ],

and x1 is connected to ∞ without edges from B(y, k). Moreover denote P0 one of theshortest paths connecting x0 to y and P1 one of the shortest paths connecting x1 to y.

Then, using the same type of arguments as in the proof of Lemma 8.2, we get forP = P0 ∪ P1 ∪ y + e, e ∈ ν, on ω such that L(i)

y = k,

E[1I(ωy,A)(Z(i)

y )C28p(i)z P

ωz,∅y,A0 [Tz < T


∗ (y)R(i)∗ (z) | Liy = k

]≤γ1k

γ2eγ3keγ4((y−z)·~)+E[10⇔ y ⇔∞P ωz,e

z [Tx0 = T∂B∗(y,k) < τδ]

× P ωz,∅y,A0 [Tz < T∂B(y,k) ∧ τδ]R(0)

∗ (y)R(0)∗ (z) | P ∈ ω

]≤γ1k

γ222k+4d+2eγ3keγ4((y−z)·~)+E[1P ∈ ω10⇔ y ⇔∞

× P ωz,e

z [Tx0 = T∂B∗(y,k) < τδ]Pωz,∅y,A0 [Tz < T∂B(y,k) ∧ τδ]R(0)

∗ (y)R(0)∗ (z)

].

Now on ω such that P ∈ ω, we have

Pωz,∅y,A0 [Tz < T

∂B(i)y∧ τδ] = P

ωz,∅y,∅

0 [Tz < T∂B

(i)y∧ τδ] ≤ P

ωz,∅y,∅

0 [Tz < τδ],

andP ωz,e

z [Tx0 = T∂B∗(y,k) < τδ](δκ0)k ≤ P ωz,∅

z+e+[Ty < τδ].

Since we also have 1P ∈ ω10⇔ y ⇔∞ ≤ 1I(ωz,∅) so that,

1P ∈ ω10⇔ y ⇔∞pωz,ez (y, k)Pωz,∅y,A0 [Tz < T∂B(y,k) ∧ τδ]

×R(0)∗ (y)R(0)

∗ (z)

150

9. AN INCREASING SPEED

≤γ5kγ6eγ7k1I(ωy,∅)Gωz,∅

y,∅δ (0, z)G

ωz,ey,∅δ (z + e+, y),

and the results follows by integration over the values of L(i)y .

Now, as we did to obtain the continuity of the speed, we need to show that thecontribution due to the local modications of the environment has a bounded eect.Hence we want to prove that the expectations appearing in Lemma 8.2 and Lemma 8.3are nite for ε small enough. This is proved using the following lemma.

Lemma 8.4. For ε9 small enough and any ε < ε9 we have

E[(L(3)y )C30+C34e(C31+C35)L

(3)y ] < C37,

where C37 depends only on d and `.

Démonstration. Let us give an upper bound on the tail of L(3)y , we have

P[L(3)y ≥ n] ≤ P[L(3)

y ≥ n, L(2)y ≤ n/(2C8)]

+ P[L(2)y ≥ n/(2C8), L(1)

y ≤ n/(2C8)2]

+ P[L(1)y ≥ n/(2C8)2],

and recalling Proposition 5.2 and Proposition 5.3 we get for A = B(x, r)

P1−ε[LA(ω) ∨ LA(ωz,e) ≥ n+ C8r] ≤ 2C9rdnα(ε)n,

so that we may use the second property of Proposition 4.1

P[L(3)y ≥ n] ≤ 6C9

( n

2C8

)dnα(ε)f(n),

where f(n) = (n/(2C8)2 − C8) and α(ε) can be arbitrarily small if we take ε smallenough. The result follows easily.

Now, Proposition 7.2 follows from the decomposition obtained at (8.27) and (8.28),the decorrelation part is handled by Lemma 8.2, Lemma 8.3 and the control on themultiplicative term appearing in these lemmas is upper bounded by Lemma 8.4 for εsmall enough.

9 An increasing speed

We want to prove Proposition 2.1 and show that the walk slows down when wepercolate, i.e. v`(1) · v′`(1) > 0 under certain conditions. We recall Je = (Gω0(0, 0) −Gω0(e, 0)) > 0 and we introduce Jee = (Gω0,e

0 (0, 0)−Gω0,e0 (e, 0)) > 0.

151


We use (7.2) to prove that

Gω0,e0 (0, 0) =Gω0(0, 0) +Gω0(0, 0)

∑e′∈ν

(pe(e′)− p∅(e′))Gω0,e0 (e′, 0)

+Gω0(0, e)∑e′∈ν

(p−e(e′)− p∅(e′))Gω0,e0 (e+ e′, 0),

and

Gω0,e0 (e, 0) =Gω0(e, 0) +Gω0(e, 0)

∑e′∈ν

(pe(e′)− p∅(e′))Gω0,e0 (e′, 0)

+Gω0(e, e)∑e′∈ν

(p−e(e′)− p∅(e′))Gω0,e0 (e+ e′, 0).

Now, recalling the proof of Lemma 7.2 (in particular (7.20) and (7.21)), noticing therelations,Gω0(e, e) = Gω0(0, 0) and by reversibilityGω0(e, 0) = (πω0(0)/πω0(e))Gω0(0, e) =(c(e)/c(−e))Gω0(0, e), we get

Jee =Je +Gω0(0, 0)[p(e)(Gω0,e

0 (0, 0)− 1)− p(e)Gω0,e0 (e, 0)

− (p(−e)Gω0,e0 (e, 0)− p(−e)Gω0,e

0 (0, 0))]

+Gω0(e, 0)[(c(e)/c(−e))(p(−e)Gω0,e

0 (e, 0)− p(−e)Gω0,e0 (0, 0))

− (p(e)(Gω0,e0 (0, 0)− 1)− p(e)Gω0,e

0 (e, 0))],

which, recalling p(e)c(−e) = p(−e)c(e), means that

Jee = Je +Gω0(0, 0)((p(e) + p(−e))Jee − p(e)) +Gω0(e, 0)(−2p(e)Jee + p(e)).

Now rewriting p(e)Gω0(e, 0) = p(−e)Gω0(0, e) = p(−e)Gω0(−e, 0), we get

Jee = Je + p(e)JeJee + p(−e)J−eJee − p(e)Je,

i.e.

Jee =(1− p(e))Je

1− p(e)Je − p(−e)J−e .

In order to obtain the alternative form of the derivative we only need to use that1− p(e) = πe/π∅ and

πe(d∅ − de) = πe(−∑e′ 6=e

c(e′)c(e)

π∅πee′ +

c(e)

π∅e)

= c(e)(e− d∅),

which proves the rst part of Proposition 2.1.

152

9. AN INCREASING SPEED

Now, we need to show that this derivative is in the same direction as v`(1), for thislet us rst notice that

1− p(e)Je − p(−e)J−e=1−Gω0(0, 0)(p(e)P ω0

e [T+0 =∞] + p(−e)P ω0

−e [T+0 =∞]) > 0,

since Gω0(0, 0) = P ω00 [T+

0 = ∞] =∑

e′∈ν p(e′)P ω0

e′ [T+0 = ∞]. Moreover the quantity in

the previous display is the same for e and −e.Now, x e ∈ ν such that e · d∅ > 0, we will show that the contribution of the terms

e and −e together are in the same direction as d∅. In fact it is

H(|e|) := (d∅ · e)[ p(e)Je + p(−e)J−e

1− p(e)Je − p(−e)J−e e−p(e)Je − p(−e)J−e

1− p(e)Je − p(−e)J−ed∅],

and since β(|e|) =: (d∅ · e)/(1− p(e)Je − p(−e)J−e) > 0 we get

H(|e|) · d∅ = β(|e|)[(p(e)Je + p(−e)J−e)(d∅ · e)− (p(e)Je − p(−e)J−e)(d∅ · d∅)]> 0,

if we suppose that

for i = 1, . . . , d such that d∅ · e(i) > 0, d∅ · e(i) ≥ ||d∅||2 .

Finally v`(1) · v′`(1) =∑d

i=1 H(∣∣e(i)

∣∣) · d∅ > 0, so that Proposition 2.1 is proved.

Acknowledgements

I would like to thank my advisor Christophe sabot for suggesting this problem andfor his support.

I also would like to thank Pierre mathieu for useful discussions and comments.My research was supported by the A.N.R. MEMEMO.

153


154

6On slowdown and speedup of transient

random walks in random environment

We consider one-dimensional random walks in random environment which are tran-sient to the right. Our main interest is in the study of the sub-ballistic regime, where attime n the particle is typically at a distance of order O(nκ) from the origin, κ ∈ (0, 1).We investigate the probabilities of moderate deviations from this behaviour. Specically,we are interested in quenched and annealed probabilities of slowdown (at time n, theparticle is at a distance of order O(nν0) from the origin, ν0 ∈ (0, κ)), and speedup (attime n, the particle is at a distance of order nν1 from the origin, ν1 ∈ (κ, 1)), for thecurrent location of the particle and for the hitting times. Also, we study probabilities ofbacktracking : at time n, the particle is located around (−nν), thus making an unusualexcursion to the left. For the slowdown, our results are valid in the ballistic case as well.

The material of this chapter is a joint work with N. Gantert and S. Popov and hasbeen accepted for publication in Probability Theory and Related Fields, see [39].

CHAPITRE 6. SLOWDOWN AND SPEEDUP OF TRANSIENT RWRE

1 Introduction and results

Let ω := (ωi, i ∈ Z) be a family of i.i.d. random variables taking values in (0, 1).Denote by P the distribution of ω and by E the corresponding expectation. After choos-ing an environment ω at random according to the law P, we dene the random walkin random environment (usually abbreviated as RWRE) as a nearest-neighbour randomwalk on Z with transition probabilities given by ω : (Xn, n ≥ 0) is the Markov chainsatisfying X0 = z and for n ≥ 0,

P zω [Xn+1 = x+ 1 | Xn = x] = ωx,

P zω [Xn+1 = x− 1 | Xn = x] = 1− ωx.

As usual, P zω is called the quenched law of (Xn, n ≥ 0) starting from X0 = z, and

we denote by Ezω the corresponding quenched expectation. Also, we denote by Pz the

semi-direct product P × P zω and by Ez the expectation with respect to Pz ; Pz and Ez

are called the annealed probability and expectation. When z = 0, we write simply Pω,Eω, P, E.

In this paper we will also consider RWRE on Z+, with reection to the right at theorigin. This RWRE can be dened as above, in the environment ω given by

ωi =

ωi, i 6= 0,

1, i = 0

(provided, of course, that the starting point is nonnegative). We then write P zω , E

zω for

the quenched probability and expectation in the case of RWRE reected at the origin,Pz and Ez for the annealed probability and expectation, keeping the simplied notationPω, Eω, P, E for the RWRE starting at the origin.

For all i ∈ Z, let us introduce

ρi :=1− ωiωi

.

Throughout this paper, we assume that

E[ln ρ0] < 0, (1.1)

which implies (cf. [95]) that limn→∞Xn = +∞ Pω-a.s. for P-a.a. ω, so that the RWREis transient to the right (or simply transient, in the case of RWRE with reection at theorigin).

We refer to [104] for a general overview of results on RWRE. In the following wealways work under the assumption that

there exists a unique κ > 0, such that E[ρκ0 ] = 1 and E[ρκ0 ln+ ρ0] <∞. (1.2)

156

1. INTRODUCTION AND RESULTS

This constant plays a central role for RWRE, in particular when it exists, its valueseparates the ballistic from the sub-ballistic regime :

κ > 1 if and only ifXn

n→ v > 0, P-a.s.

We refer to the case κ > 1 as the ballistic regime and to the case κ ≤ 1 as the sub-ballistic regime. In this paper we mainly consider the case where the RWRE is transient(to the right) and sub-ballistic, i.e. the asymptotic speed is equal to 0. The followingresult was proved in [59] and partially rened in [35] :

Theorem 1.1. Let ω := (ωi, i ∈ Z) be a family of independent and identically dis-tributed random variables such that

(i) −∞ ≤ E[ln ρ0] < 0,(ii) there exists 0 < κ ≤ 1 for which E [ρκ0 ] = 1 and E

[ρκ0 ln+ ρ0

]<∞,

(iii) the distribution of ln ρ0 is non-lattice.Then, if κ < 1, we have

Xn

nκlaw−→ C1

(1

Scaκ

)κ,

wherelaw−→ stands for convergence in distribution with respect to the annealed law P, C1

is a positive constant and Scaκ is the completely asymmetric stable law of index κ. Ifκ = 1, we have

Xn

n/ lnn

law−→ C21

Sca1

.

In the quenched case, the limiting behaviour is more complicated, as discussed in [79].However, one still can say that at time n the particle is typically at distance roughly nκ

from the origin, since the weaker result limn→∞ lnXn/ lnn = κ, P-a.s., is still valid1.Besides the results about the location of the particle at time n, we are interested

also in the rst hitting times of certain regions in space. For any set A ⊂ Z, dene :

TA := minn ≥ 0 : Xn ∈ A.

To simplify the notations, for one-point sets we write Ta := Ta. In the case where a isnot an integer, the notation Ta will correspond to Tbac.

In this paper we investigate the following types of unusual behaviour of the randomwalk :

slowdown, which means that at time n the particle is around nν0 , ν0 < 1 ∧ κ, sothat the particle goes to the right much slower than it typically does ;

1apparently, this result is folklore, at least we were unable to nd a precise reference in the literature.

Anyhow, note that it is straightforward to obtain this result from Theorems 1.2 and 1.5

157


backtracking, that is, at time n the particle is found around (−nν), thus performingan unlikely excursion to the left instead of going to the right (this is, of course,only for RWRE without reection) ;

speedup, which means that the particle is going to the right faster than it should(but still with sublinear speed) : at time n the particle is around nν1 , κ < ν1 < 1(this is possible only for κ < 1).

We refer to all of the above as moderate deviations, even for the slowdown in theballistic case κ > 1. Indeed, in the latter case the deviation from the typical position islinear in time, but we have that the large deviation rate function I satises I(0) = 0,and the known large deviation results only tell us that slowdown probabilities decayslower than exponentially in n (see, for instance, [20]).

We mention here that in the literature one can nd some results on moderate devi-ations for the case of recurrent RWRE (often referred to as RWRE in Sinai's regime),see [21, 22], and also [51] for the continuous space and time version.

Now, we state the results we are going to prove in this paper. In addition to (1.2),we will use the following weak integrability hypothesis :

there exists ε0 > 0 such that E[ρ−ε00 ] <∞. (1.3)

First, we discuss the results about quenched slowdown probabilities. It turns outthat the quenched slowdown probabilities behave dierently depending on whether oneconsiders RWRE with or without reection at the origin. Also, it matters which of thefollowing two events is considered : (i) the position of the particle at time n is at mostnν , ν < κ (i.e., the event Xn < nν), or (ii) the hitting time of nν is greater than n (i.e.,the event Tnν > n). Here we prove that in all these cases the quenched probabilityof slowdown is roughly e−n

β, where β = 1 − ν

κfor the hitting time slowdown in the

reected case, and β = (1− νκ) ∧ κ

κ+1in the other cases. More precisely, we have

Theorem 1.2. Slowdown, quenched Suppose that (1.1), (1.2) and (1.3) hold. Forν ∈ (0, 1 ∧ κ) the quenched slowdown probabilities behave in the following way. For thereected RWRE,

limn→∞

ln(− lnPω[Tnν > n])

lnn= 1− ν

κ, P-a.s., (1.4)

limn→∞

ln(− lnPω[Xn < nν ])

lnn=(

1− ν

κ

)∧ κ

κ+ 1, P-a.s. (1.5)

For the RWRE without reection, we obtain

limn→∞


lnn=(

1− ν

κ

)∧ κ

κ+ 1, P-a.s., (1.6)

limn→∞


lnn=(

1− ν

κ

)∧ κ

κ+ 1, P-a.s. (1.7)

158

1. INTRODUCTION AND RESULTS

For a heuristical explanation of the reason for the dierent behaviours of the quenchedslowdown probabilities we refer to the beginning of Section 6.

For the annealed slowdown probabilities, we obtain that there is no dierence be-tween reecting/nonreecting cases (at least on the level of precision we are work-ing here) and also it does not matter which one of the slowdown events Tnν > n,Xn < nν one considers. In all these cases, the annealed probability of slowdown de-cays polynomially, roughly as n−(κ−ν) :

Theorem 1.3. Slowdown, annealed Suppose that (1.1), (1.2) and (1.3) hold. Forν ∈ (0, 1 ∧ κ),

limn→∞

ln P[Xn < nν ]

lnn= lim

n→∞

ln P[Tnν > n]

lnn= −(κ− ν). (1.8)

The same result holds if one changes P to P in (1.8).

In the case of RWRE on Z (i.e., without reection at the origin) there is anotherkind of untypically slow escape to the right. Namely, before going to +∞, the particlecan make an untypically big excursion to the left of the origin. While it is easy tocontrol the distribution of the leftmost site touched by this excursion (e.g., by meansof the formula (2.8) below), it is interesting to study the probability that at time n theparticle is far away to the left of the origin :

Theorem 1.4. Backtracking Suppose that (1.1), (1.2) and (1.3) hold. For ν ∈ (0, 1),we have

limn→∞

ln(− lnPω[Xn < −nν ])lnn

= ν ∨ κ

κ+ 1, P-a.s. (1.9)

limn→∞

ln(− ln P[Xn < −nν ])lnn

= ν, (1.10)

and

limn→∞

ln(− ln P[T−nν < n])

lnn= lim

n→∞

ln(− lnPω[T−nν < n])

lnn= ν P-a.s. (1.11)

Another kind of deviation from the typical behaviour is the speedup of the particle,i.e., at time n the particle is at a distance larger than nκ from the origin (here weof course assume that κ < 1). There are results in the literature that cover the largedeviations case, i.e., the case when at time n the particle is at distance O(n) fromthe origin, see e.g. Section 2.3 of [104], or [20]. In this paper we are interested in theprobabilities of moderate speedup : the displacement of the particle is sublinear, butstill bigger than in the typical case. Namely, we show that the quenched probabilitythat Xn is of order nν , κ < ν < 1, is roughly e−n

β, where β = ν−κ

1−κ . It is remarkablethat the annealed probability is roughly of the same order. More precisely, we are ableto prove the following result :

159


Theorem 1.5. Speedup Suppose that (1.1), (1.2) and (1.3) hold. For ν ∈ (κ, 1) wecan control the probabilities of the moderate speedup in the following way :

limn→∞

ln(− lnPω[Xn > nν ])

lnn= lim

n→∞

ln(− lnPω[Tnν < n])

lnn=ν − κ1− κ , P-a.s., (1.12)

and

limn→∞

ln(− ln P[Xn > nν ])

lnn= lim

n→∞

ln(− ln P[Tnν < n])

lnn=ν − κ1− κ . (1.13)

The same result holds for the RWRE with reection at the origin.

For the case κ ∈ (0, 1), the quenched moderate deviations for the random walk on Zare well summed up by the plot of the following function on Figure 6.1 :

f(ν) =

limn→∞ ln(− lnPω[Xn < −n−ν ])/ lnn, if ν ∈ (−1, 0],limn→∞ ln(− lnPω[Xn < nν ])/ lnn, if ν ∈ (0, κ),limn→∞ ln(− lnPω[Xn > nν ])/ lnn, if ν ∈ [κ, 1).

−1 − κκ+1

0 κκ+1

κ 1

1f(ν)

ν

κκ+1

Fig. 6.1 The plot of f(ν), −1 < ν < 1

The rest of this paper is organized in the following way. In Section 2 we give the(standard) denition of the potential and the reversible measure for the RWRE. We thendecompose the environment into a sequence of valleys. In this decomposition the valleysdo not only depend on the environment but the construction is time-dependent. Also,we derive some basic facts about the valleys needed later. In Section 3 we mainly studythe properties of that sequence of valleys. In Section 4, we recall some results concerningthe spectral properties of RWRE restricted to a nite interval, and then obtain somebounds on the probability of connement in a valley. In Section 5 we dene the induced

160

2. MORE NOTATIONS AND SOME BASIC FACTS

random walk whose state is the current valley (more precisely, the last visited boundarybetween two neighbouring valleys) where the particle is located. Theorems 1.2, 1.3, 1.4,1.5 are proved in Sections 6, 7, 8, 9 respectively. We denote by γ, γ0, γ1, γ2, γ3, . . . theimportant constants (those that can be used far away from the place where they appearfor the rst time), and by C1, C2, C3, . . . the local ones (those that are used only in asmall neighbourhood of the place where they appear for the rst time), restarting thenumeration at the beginning of each section in the latter case. All these constants areeither universal or depend only on the law of the environment.

2 More notations and some basic facts

An important ingredient of our proofs is the analysis of the potential associatedwith the environment, which was introduced by Sinai in [94]. The potential, denoted byV = (V (x), x ∈ Z), is a function of the environment ω. It is dened in the followingway :

V (x) :=

∑x

i=1 ln ρi, if x ≥ 1,

0, if x = 0,

−∑0i=x+1 ln ρi, if x ≤ −1,

so it is a random walk with negative drift, because E[ln ρ0] < 0. This notation is extendedon R by V (x) := V (bxc). We also dene a reversible measure

π(x) := e−V (x) + e−V (x−1), for x ∈ Z, (2.1)

(one easily veries that ωxπ(x) = (1 − ωx+1)π(x + 1) for all x). We will also use thenotation π([x, y]) =

∑byci=bxc−1 π(i), for x < y two real numbers.

The function V (·) enables us to dene the valleys, parts of the environment whichacts as traps for the random walk. The valleys are responsible for the sub-ballisticbehaviour and hence play a central role for slowdown and speedup phenomena.

We dene by induction the following environment dependent sequence (Ki(n))i≥0 by

K0(n) =− n,Ki+1(n) = min

j ≥ Ki(n) : V (Ki(n))− min

k∈[Ki(n),j]V (k) ≥ 3

1 ∧ κ lnn,

V (j) = maxk≥j

V (k).

The dependence with respect to n will be frequently omitted to ease the notations. Theportion of the environment [Ki, Ki+1) is called the i-th valley, and we will prove thatfor n large enough the valleys are descending in the sense that V (Ki+1) < V (Ki) for alli ∈ [0, n]. We associate to the i-th valley the bottom point

bi = infx ∈ [Ki, Ki+1) : V (x) = min

y∈[Ki,Ki+1)V (y)

,

161


x

V (x)

−n = K0(n) 0

Ki(n) bi Ki+1(n)

Hi

≥ 31∧κ lnn

Fig. 6.2 On the denition of the sequence of valleys

and the depth

Hi = maxx∈[Ki,Ki+1)

(max

y∈[x,Ki+1)V (y)− min

y∈[Ki,x)V (y)

)= max

Ki(n)≤j<k<Ki+1(n)

(V (k)− V (j)

),

see Figure 6.2.Let us denote

Nn(m,m′) = i ≥ 1 : [Ki, Ki+1) ∩ [bmc, bm′c) 6= ∅ (2.2)

and again we will often omit the index n. Let us emphasize that we do not include thevalley of index 0, which is dierent from the others because of border issues.

The valleys for i ≥ 1 are non-overlapping parts of Z, for any value of n. Moreover thepotential in the valleys are i.i.d. up to space-shift, in the sense that for any n and i ≥ 1 thesequence of vectors of random length

(V (j)− V (Ki+1(n)), j = Ki(n), . . . , Ki+1(n)− 1

),

i ≥ 1, is i.i.d.We introduce the two following indices which will be used regularly

i0 = cardN(−n, 0) and i1 = cardN(−n, nν). (2.3)

162

3. ESTIMATES ON THE ENVIRONMENT

To carry over the proofs easily to the reected case, we introduce the followingnotation

Ki0 = 0 and Ki = Ki for i ≥ i0. (2.4)

We can estimate the depth of the valleys using a result of renewal theory whichconcerns the maximum of random walks with negative drift. We refer to [37] for adetailed introduction to renewal theory. Denoting S = maxi≥0 V (i), under assumptions(1.1), (1.2) and (iii) of Theorem 1.1, we have

P[S > h] ∼ CF e−κh, h→∞, (2.5)

which is a result due to Feller which can be found in this form in [54].If (iii) in Theorem 1.1 fails, ln ρ0 is concentrated on λZ for some λ > 0, so that V (·)

is a Markov chain with i.i.d. increments of law ln ρi. In this case, under our assump-tions (1.1) and (1.2) we can use a result in [96] (p. 218) stating the discrete version ofthe previous equation. In the case of an aperiodic Markov chain we have

P[S ≥ nλ] ∼ C ′F e−κλn, n→∞, (2.6)

and in the general case we obtain similar asymptotics by noticing that (V (nd + k))n≥0

is aperiodic for k ∈ [0, d − 1] and d the period of V (·) (which is well dened and niteby (i) and (ii)).

Hence we can easily deduce from the assumptions (1.1) and (1.2) and equations (2.5)and (2.6) that

P[S > h] = Θ(e−κh), (2.7)

where f(n) = Θ(g(n)) means that f(n) = O(g(n)) and g(n) = O(f(n)).Let us recall also the following basic fact. For any integers a < x < b, the (quenched)

probability for RWRE to reach b before a starting from x can be easily computed :

P xω [Tb < Ta] =

∑x−1y=a e

V (y)∑b−1y=a e

V (y), (2.8)

see e.g. Lemma 1 in [94] or formula (2.1.4) in [104].

3 Estimates on the environment

Let us introduce the event

A(n) =

maxi≤2n

(Ki+1 −Ki) ≤ (lnn)2. (3.1)

The following lemma shows that the valleys are not very wide.

163


Lemma 3.1. We have

P[A(n)c] = O( 1

n2

).

Démonstration. We have

P[A(n)c] = P[maxi≤2n

(Ki+1 −Ki) > (lnn)2]

≤ 2nP[K2 −K1 > (lnn)2] + P[K1 > (lnn)2], (3.2)

where

K1(n) = minj ≥ 0 : − min

k∈[0,j]V (k) ≥ 3

1 ∧ κ lnn, V (j) = maxk≥j

V (k).

Now

P[K2 −K1 > (lnn)2] = P[K1 > (lnn)2 | maxi≥0

V (i) ≤ 0]

≤ P[K1 > (lnn)2]

P[maxi≥0 V (i) ≤ 0],

where P[maxi≥0 V (i) ≤ 0] > 0 since E[ln ρ0] < 0. Choose ` such that ε0` > 3(1 ∧ κ),with ε0 from (1.3). Note that if V ((lnn)2) ≤ −3+3`

1∧κ lnn, minj≤(lnn)2 (V (j)− V (j − 1)) ≥− `

1∧κ lnn and maxj≥(lnn)2 V (j)− V ((lnn)2) ≤ 31∧κ lnn, then the set

i ∈ [0, (lnn)2], V (i) ∈(− 3

1 ∧ κ lnn,−3 + 2`

1 ∧ κ lnn

)is non-empty. Moreover its largest element m is such that maxj≥m V (j) = V (m), hencewe have K1 ≤ (lnn)2. This yields

P[K1 > (lnn)2] ≤ P[V ((lnn)2) > −3 + 3`

1 ∧ κ lnn (3.3)

or minj≤(lnn)2

(V (j)− V (j − 1)) < − `

1 ∧ κ lnn

or maxj≥(lnn)2

V (j)− V ((lnn)2) >3

1 ∧ κ lnn].

Using (2.7), we obtain

P[

maxj≥(lnn)2

V (j)− V ((lnn)2) >3

1 ∧ κ lnn]

= O(n−3). (3.4)

Furthermore, using Chebyshev's inequality and (1.3) we get

P[

minj≤(lnn)2

(V (j)− V (j − 1)) < − `

1 ∧ κ lnn]

164


≤ (lnn)2P

[ln ρ0 < − `

1 ∧ κ lnn

]≤ (lnn)2P

[ρ−ε00 > exp

(ε0

`

1 ∧ κ lnn

)]≤ (lnn)2E[ρ−ε00 ]n−ε0`/(1∧κ)

= o(n−3) . (3.5)

Now, since V (·) is a sum of i.i.d. random variables with exponential moments by theassumptions (1.2) and (1.3), we can use large deviations techniques to get

P[V ((lnn)2) > −C1 lnn] ≤ P[∣∣V ((lnn)2)− E[V (1)](lnn)2

∣∣ > C2(lnn)2]

(3.6)

≤ exp(−C3(lnn)2)

= o(n−3),

since E[V (1)] = E[ln ρ0] ∈ (−∞, 0). Putting together (3.2), (3.3), (3.4), (3.5) and (3.6)we obtain the result.

Consider a ∈ [0, ν), and dene the event

B(n, ν, a)c =

cardi ∈ Nn(−nν , nν) : Hi ≥ a

κlnn+ ln lnn

≥ nν−a

.

The following lemma will tell us that asymptotically, between levels −nν and nν

there are at most nν−a valleys of depth greater than (a/κ) lnn+ ln lnn.

Lemma 3.2. For any a ∈ [0, ν), we have

P[B(n, ν, a)c] = O(n−2).

Démonstration. We have easily that (≺ means stochastically dominated)

cardi ≤ Nn(−nν , nν) : Hi ≥ a

κlnn+ ln lnn

≺ Bin

(2bnνc+ 2,P

[S ≥ a

κlnn+ ln lnn

]),

since we have at most 2bnνc + 2 integers on the right of which we need an increase ofpotential of (a/κ) lnn+ ln lnn to create a valley of sucient depth.

Using (2.7), we have

P[S ≥ a

κlnn+ ln lnn

]= O

( n−a

(lnn)κ

).

Now, using Chebyshev's exponential inequality, we can write

P[Bin(

2bnνc+ 2,P[S ≥ a

κlnn+ ln lnn

])≥ nν−a

]≤ C4 exp(−nν−a) exp(C5n

ν−a(lnn)−κ),

and, since ν > a, the result follows.

165


We introduce for m ∈ Z+ the following event, which, by Lemma 3.2, has probabilityconverging to 1,

B′(n, ν,m) =m−1⋂k=1

B(n, ν, kν/m). (3.7)

Also, set

G(n)c =

maxk≥n

(V (k)− V (n)) ≥ 1

κ(lnn+ 2 ln lnn)

⋃

maxk≥−n

(V (k)− V (−n)) ≥ 1

κ(lnn+ 2 ln lnn)

.

Lemma 3.3. We have

P[G(n)c] = O( 1

n(lnn)2

).

Démonstration. This is a direct consequence of (2.7).

We now show that Lemma 3.3 implies that asymptotically, in the interval [−n, n],the deepest valley we can nd has depth lower than 1

κ(lnn+ 2 ln lnn). Let

G1(n) =

maxi∈[−n,n]

maxk≥i

(V (k)− V (i)) ≤ 1

κ(lnn+ 2 ln lnn)

. (3.8)

Lemma 3.4. For P-almost all ω, there is N = N(ω) such that ω ∈ G1(n) for n ≥ N .

Démonstration. By symmetry, it suces to give the proof for

G2(n) =

maxi∈[0,n]

maxk≥i

(V (k)− V (i)) ≤ 1

κ(lnn+ 2 ln lnn)

(3.9)

instead of G1(n). Let

n0 := minj ≥ 0 : max

k≥i(V (k)− V (i)) ≤ 1

κ(ln i+ 2 ln ln i), ∀i ≥ j

and

K = max0≤i≤n0

maxk≥i

(V (k)− V (i)).

Due to Lemma 3.3, n0 is nite P-almost surely. Now, take N large enough such thatN ≥ n0 and

1

κ(lnN + 2 ln lnN) ≥ K.

Then for n ≥ N , let ` ∈ [0, n] be such that maxi∈[0,n] maxk≥i(V (k)−V (i)) = maxk≥`(V (k)−V (`)). We have either ` ≤ n0 and then maxk≥`(V (k)−V (`)) ≤ K by the denition of K,or ` > n0 and then, by the denition of n0, maxk≥`(V (k)− V (`)) ≤ 1

κ(ln `+ 2 ln ln `) ≤

1κ(lnn+ 2 ln lnn).

166


Let us dene

D(n)c =

maxi∈[0,n]

maxk≥i

(V (k)− V (i)) ≤ 1

κ(lnn− 4 ln lnn)

⋃

maxi∈[−n,0]

maxk≥i

(V (k)− V (i)) ≤ 1

κ(lnn− 4 ln lnn)

.

Lemma 3.5. We haveP[D(n)c] = O(n−2).

Démonstration. First, we notice that

P[D(n)c] ≤ 2P[

maxi∈[0, n

b(lnn)2c]

maxk≤(lnn)2

V (i(lnn)2 + k)− V (i(lnn)2)

≤ 1

κ(lnn− 4 ln lnn)

]+ P[A(n)c],

where P[A(n)c] = O(n−2) by Lemma 3.1.Let us introduce

D(1)(n) =

maxk>b(lnn)2c

V (k)− V (0) ≥ 1

κ(lnn− 4 ln lnn)

,

then we have

P[D(1)(n)] ≤P[maxk≥0

V (k)− V (0) >1

κ(lnn− 4 ln lnn)

]+ P

[maxk≥0

V (k)− V (0) 6= maxk≤(lnn)2

V (k)− V (0)]

= Θ((lnn)4

n

),

using a reasoning similar to the proof of Lemma 3.1 (cf. equations (3.3) and (3.4)) toshow that the second term is at most O(n−2).

So, we obtain for n large enough

P[D(n)c] ≤ 2(

1− C6(lnn)4

n

)n/(lnn)2

≤ 2 exp(−C7(lnn)2

),

hence the result.

Finally, let us introduce

F (n) =

mini∈[−n,n]

(1− ωi) > n−3/ε0.

Lemma 3.6. We have

P[F (n)c] = O( 1

n2

).

167


Démonstration. We notice that 1 − ωi ≥ min(1/2, ρi/2), so that it is enough to provethat P[ρi < 2n−3/ε0 ] = O(n−3) which is a consequence of (1.3), since by Chebyshev'sinequality

P[ρ−1i >

n3/ε0

2

]≤ 2ε0E[ρ−ε00 ]

n3.

Using the Borel-Cantelli Lemma one can obtain that for P-almost all ω and n largeenough, we have ω ∈ A(n)∩B′(n, ν,m)∩G1(n)∩D(n)∩F (n). That is, the width of thevalleys is lower than (lnn)2, their depth lower than (lnn + 2 ln lnn)/κ, we can controlthe number of valleys deeper than a

κlnn − ln lnn, and there is at least one valley of

depth (lnn− 4 ln lnn)/κ.Due to the denition of the valleys, the potential goes down at least by 3

1∧κ lnn in avalley and on G1(n) the biggest increase of potential is lower than 1

κ(lnn+ 2 ln lnn) for

all valleys in [−n, n]. In particular, on G1(n), (V (Ki))i≤2n is a decreasing sequence andwe have

V (bi+1) ≤ V (bi)− 3

1 ∧ κ lnn+1

κ(lnn+ 2 ln lnn)

≤ V (bi)− 2

1 ∧ κ lnn+2

κln lnn

implying using (2.1) that for all valleys in [−n, n],

π(bi) ≤ 2e−V (bi) ≤ 2(lnn)2/κ

n2/(1∧κ)π(bi+1) ≤ 1

2π(bi+1). (3.10)

In a similar fashion, we can give an upper bound for V (Ki)−V (bi) on G1(n)∩F (n).We claim that on G1(n) ∩ F (n), for a constant γ0,

V (Ki)− V (Ki+1) ≤ V (Ki)− V (bi) ≤ γ0 lnn. (3.11)

To show (3.11), let x be the smallest integer larger than Ki such that V (x) ≤ V (Ki)−(3/(1 ∧ κ)) lnn. By denition of Ki+1 it satises V (x) ≤ V (Ki+1). But on F (n) weknow that V (x) ≥ V (Ki) − (3/(1 ∧ κ) + 3/ε0) lnn. Recalling that on G1(n) we haveV (bi) ≥ V (Ki+1)− (2/κ) lnn, we get for n large enough

V (Ki)− V (bi) ≤ V (Ki)− (V (Ki+1)− 2

κlnn)

≤ V (Ki)− (V (x)− 2

κlnn)

≤( 3

1 ∧ κ +3

ε0

+2

κ

)lnn .

168

4. BOUNDS ON THE PROBABILITY OF CONFINEMENT

4 Bounds on the probability of connement

In this section, let I = [a, c] be a nite interval of Z containing at least four pointsand let the potential V (x) be an arbitrary function dened for x ∈ [a − 1, c], withV (a− 1) = 0. This potential denes transition probabilities given by ωx = e−V (x)/π(x),x ∈ [a, c] where π(x) is dened as in (2.1) (taking V (a− 1) = 0 is no loss of generalitysince the transition probabilities remain the same if we replace V (x) by V (x) + c, ∀x).We denote by X the Markov chain restricted on I in the following way : the transitionprobability ωa from a to a+ 1 is dened as above, and with probability 1−ωa the walkjust stays in a ; in the same way, we dene the reection at the other border c. Wedenote

H+ = maxx∈[a,c]

(maxy∈[x,c]

V (y)− miny∈[a,x)

V (y)),

H− = maxx∈[a,c]

(maxy∈[a,x]

V (y)− miny∈(x,c]

V (y)),

andH = H+ ∧H−.

Let us denote also byM = max

y∈[a,c]V (y)− min

y∈[a,c]V (y)

the maximal dierence between the values of the potential in the interval [a, c]. Also,we set

f =

c, if H = H+,

a, otherwise.

To avoid confusion, let us mention that the results of this section (Propositions 4.1,4.2, 4.3) hold for both the unrestricted and restricted random walks (as long as thestarting point belongs to I). First, we prove the following

Proposition 4.1. There exists γ1 > 0, such that for all u ≥ 1

maxx∈I

P xω

[ Ta,c

γ1(c− a)3((c− a) + M)eH> u

]≤ max

x∈IP xω

[ Tf

γ1(c− a)3((c− a) + M)eH> u

]≤ e−u.

Démonstration. The rst inequality is trivial, we only need to prove the second one. Inthe following we will suppose that H = H+ (so that f = c), otherwise we can apply the

169


same argument by inverting the space. We denote by b the leftmost point in the interval[a, c] with minimal potential.

We extend the Markov chain on the interval I to a Markov chain on the interval I ′ =[a, c+1] in the following way. Let V (c+1) := V (b), yielding ωc+1 =

(1+e−(V (c)−V (b))

)−1.

Again, with probability 1 − ωc+1, the Markov chain goes from c + 1 to c, and withprobability ωc+1, the Markov chain just stays in c+ 1.

Let us denote by Xt the continuous time version of the Markov chain on I ′ (i.e., thetransition probabilities become transition rates). The reason for considering continuoustime is the following : we are going to use spectral gap estimates, and these are bettersuited for continuous time in this context (mainly due to the fact that the discrete-timerandom walk is periodic). We dene the probability measure µ on I ′ which is reversible(and therefore invariant) for X in the following way

µ(x) = π(x)(∑y∈I′

π(y))−1

,

for all x ∈ I ′, where π is as in (2.1) with the potential dened above, satisfying V (a−1) =0 and V (c + 1) = V (b). Now, the goal is to bound the spectral gap λ(I ′) from below.We can do this using a result of [76] :

1

4BI′≤ λ(I ′) ≤ 2

BI′, (4.1)

where BI′ = mini∈I′(BI′− (i) ∧BI′

+ (i)) and

BI′

+ (i) = maxx>i

(x∑

y=i+1

(µ(y)(1− ωy))−1

)µ[x, c+ 1], i ∈ [a, c]

BI′

− (i) = maxx<i

(i−1∑y=x

(µ(y)ωy)−1

)µ[a, x], i ∈ [a+ 1, c+ 1]

and BI′+ (c+ 1) = BI′

− (a) = 0. Obviously, we have BI′ ≤ BI′− (c+ 1). Moreover, since (2.1)

implies that ωxπ(x) = e−V (x) for any x ∈ I ′, we can write

BI′

− (c+ 1) = maxx≤c

( c∑y=x

1

ωyπ(y)

)( x∑y=a

π(y))

= maxx≤c

( c∑y=x

eV (y))( x∑

y=a

(e−V (y) + e−V (y−1)))

≤ 2 maxx≤c

( c∑y=x

eV (y))( x∑

y=a

e−V (y))

170

4. BOUNDS ON THE PROBABILITY OF CONFINEMENT

≤ 2(c− a)2eH .

This yields

λ(I ′) ≥ 1

8(c− a)2eH.

Using Corollary 2.1.5 of [91], we obtain that for x, y ∈ I ′ and s > 0∣∣∣P xω [Xs = y]− µ(y)

∣∣∣ ≤ (µ(y)

µ(x)

)1/2

exp(−λ(I ′)s).

We apply this formula for y = c + 1. Note that, using (2.1), we obtain that (µ(c +

1)/µ(x))1/2 ≤ √2eM/2 for any x ∈ (a, c). So, for s := C1(c−a)2((c−a)+M)eH , if C1 > 4is chosen large enough∣∣∣P x

ω [Xs = c+ 1]− µ(c+ 1)∣∣∣ ≤ √2e−C1(c−a)/8 <

1

8(c− a),

and, since µ(c+ 1) ≥ 1/2(c+ 1− a) ≥ 1/(4(c− a)), we obtain

minx∈I′

P xω [Xs = c+ 1] ≥ 1

8(c− a).

Let us divide [0, t] into N := bt/sc subintervals. Using the above inequality andMarkov's property we obtain (T stands for the hitting time with respect to X)

P xω [Tc > t] ≤ P x

ω [Tc+1 > t]

≤ P xω [Xsk 6= c+ 1, k = 1, . . . , N ]

≤(

1− 1

8(c− a)

)N≤ exp

(− N

8(c− a)

)≤ exp

(− t

8C1(c− a)3((c− a) + M)eH

)exp( 1

8(c− a)

).

The estimates on the continuous time Markov chain transfer to discrete time. In-deed, there exists a family (ei)i≥1 of exponential random variables of parameter 1, suchthat the n-th jump of the continuous time random walk occurs at

∑ni=1 ei. These ran-

dom variables are independent of the environment and the discrete-time random walk.Moreover, P [e1 + · · ·+ en ≥ n] ≥ 1/3, for all n. So, for any t,

1

3P[Tc ≥ t] ≤ P[Tc ≥ t]P[Tc ≥ Tc] = P[Tc ≥ t, Tc ≥ Tc] ≤ P[Tc ≥ t],

171


Hence, we have for all v > 0

maxx∈I

P xω

[ Tc

8(1 + v)C1(c− a)3((c− a) + M)eH> u

]≤(

3e1/8e−vu)e−u,

for all u ≥ 0. Hence for u ≥ 1, choosing v large enough in such a way that 3 exp(18−v) ≤

1, we obtain the result with γ1 = 8C1(1 + v).

Next, we recall the following simple upper bound on hitting probabilities :

Proposition 4.2. There exists γ2 such that for any x, y and h ∈ [x, y] we have

P xω [Ty < s] ≤ γ2(1 + s)

π(h)

π(x).

Démonstration. We can adapt Lemma 3.4 of [21] (which used a uniform ellipticity con-dition). We remain in the continuous time setting and, considering the event that y is vis-ited before time s and left again at least one time unit later (on which

∫ s+1

01Xu = ydu ≥

1), we have ∫ s+1

0

P xω [Xu = y]du ≥ P x

ω [Ty < s] · P [e1 ≥ 1] (4.2)

where e1 is an exponential random variable of parameter 1. Hence

P xω [Ty < s] ≤ P x

ω [Th < s]

≤ e

∫ s+1

0

P xω [Xu = h]du

= e

∫ s+1

0

π(h)

π(x)P hω [Xu = x]du

≤ e(s+ 1)π(h)

π(x).

Again, one can easily transfer the estimates on the continuous time Markov chain todiscrete time.

Let us now introduce

H∗+ = maxx∈[a+1,c−1]

(max

y∈[x,c−1]V (y)− min

y∈[a+1,x)V (y)

),

H∗− = maxx∈[a+1,c−1]

(max

y∈[a+1,x]V (y)− min

y∈(x,c−1]V (y)

),

andH∗ = H∗+ ∧H∗−.

We obtain a lower bound on the connement probability in the following proposition.Recall that b is the leftmost point in the interval [a, c] with minimal potential.

172

5. INDUCED RANDOM WALK

Proposition 4.3. Suppose that c − 1 has maximal potential on [b, c − 1] and a hasmaximal potential on [a, b]. Then, there exists γ3 > 0, such that for all u ≥ 1

minx∈I

P xω

[γ3 ln(2(c− a))

Ta,ceH∗

≥ u]≥ 1

2(c− a)e−u,

if eH∗ ≥ 16γ2.

Démonstration. Noticing that

minb<h<c−1

π(h)

π(b)≤ 2e−H

∗+ and min

a+1<h<b

π(h)

π(b)≤ 2e−H

∗− ,

we can apply Proposition 4.2 to obtain that

for all s ≥ 1, P bω[Ta,c < s] ≤ 8γ2se

−H∗ , (4.3)

Hence for s = eH∗/(16γ2) ≥ 1, the right-hand side of the previous inequality equals 1/2.

Now, using the exit probability formula (2.8), we obtain that

minx∈I

P xω [Tb < Ta,c] ≥ (c− a)−1. (4.4)

Denoting N = dt/se, we obtain for x ∈ I,P xω [Ta,c > t] ≥ (2(c− a))−(N+1)

≥ exp(−C2t ln(2(c− a))

eH∗

)(2(c− a))−1.

We used the following reasoning in the above calculation. Start from any x ∈ (a, c),by (4.4) the particle hits b before a, c with probability at least (c − a)−1. Then,during s time units, a, c will not be hit with probability at least 1/2. After that, theparticle is found in some x′ ∈ (a, c) and at least s time units elapsed from the initialmoment. So the cost of preventing the occurrence of Ta,c during any time interval oflength s is at most (2(c− a))−1. The result follows for γ3 large enough.

Our main application of Proposition 4.1 and Proposition 4.3, will be to control theexit times of valleys, more precisely we will be able to give upper bounds on the tail ofTKi,Ki+1 and lower bounds on the tail of TKi−1,Ki+1+1 in terms of Hi.

5 Induced random walk

Let us denote (sk(n))k≥0 the sequence dened by

s0(n) = 0,

173


si+1(n) = minj ≥ si(n) : Xj ∈ Kl(n), l ≥ 0.Then, we dene Yi = Xsi , the embedded random walk with state space Kl, l ≥ 0,

enumerating the successive valleys we visit and ln(ν) = maxi : si ≤ Tnν the numbersof steps made by the embedded random walk to reach [nν ;∞). For the reected case,we will use the same notation, replacing Kl, l ≥ 0 with Kl, l ≥ 0 dened in (2.4).

Recall (2.3) and let us denote

ξν(i) = cardj ∈ [0, ln(ν)] : Yj = Ki+1, Yj+1 = Ki for i = i0 + 1, . . . , i1 − 1,

and in order to carry over the proofs to the reected case

ξν(i) = cardj ∈ [0, ln(ν)] : Yj = Ki+1, Yj+1 = Ki for i = i0 + 1, . . . , i1 − 1.

Moreover, we introduce the real time elapsed, i.e. in the clock of Xn, during the rstleft-right crossing of the i-th valley

T next(i) = TKi+1 θ(next(i))− next(i),

where θ denotes the time-shift for the random walk and

next(i) = infn ≥ 0 : Xn = Ki, TKi+1 θ(n) < TKi−1

θ(n).In this way, each time the embedded random walk backtracks, T next(i) is the time thewalk will need to make the necessary left-right crossing of the corresponding valley.Recall (2.2). Conditionally on (Yi)i≥1 we have that (dir stands for direct, and backstands for backtrack)

Tnν = Tinit + Tdir + Tback + Tleft + Tright, (5.1)

where

Tinit =

TKi0+1

, if TKi0+1< TKi0 ,

TKi0 + T next(i0) θ(TKi0 ), else,

Tleft =

cardi ≤ Tnν : Xi < K1 without reection,

∑ln(ν)j=0 1Yj = Ki0+1, Yj+1 = Ki0×(TKi θ(sj)− sj + T next(i) (TKi θ(sj))

)with reection,

Tright = Tnν θ(next∗(i1))− next∗(i1),

Tdir =

i1−1∑i=i0+1

T next(i) θ(TKi),

174

5. INDUCED RANDOM WALK

Tback =

∑i1−1i=1

∑ln(ν)j=0 1Yj = Ki+1, Yj+1 = Ki

×(TKi θ(sj)− sj + T next(i) (TKi θ(sj))

)without reection,

∑i1−1i=i0+1

∑ln(ν)j=0 1Yj = Ki+1, Yj+1 = Ki

×(TKi θ(sj)− sj + T next(i) (TKi θ(sj))

)with reection,

where next∗(i1) = infn ≥ 0 : Xn = Ki1 , Tnν θ(n) < TKi1−1θ(n). In the reected case,

replaceKi with Ki in all the above denitions except for that of Tleft. This decompositionis illustrated on Figure 6.3 for the non-reected case.

In the non-reected case, we have the following equalities in law (for each ω) :

Tinit = τ(0), (5.2)

Tright = τ(nν), (5.3)

Tdir =

i1−1∑i=i0+1

τ(0)+ (i), (5.4)

Tback =

i1−2∑i=1

(τ(1)+ (i) + τ

(1)− (i) + · · ·+ τ

(ξν(i))+ (i) + τ

(ξν(i))− (i)) (5.5)

+

ξν(i1−1)∑j=1

τ(j)+ (i1 − 1) + τ

last,(j)− ,

where τ (j)+ (i), τ (j)

− (i) and τlast,(j)− are independent sequences of i.i.d. random variables

described as follows. First, τ (j)+ (i) is a sequence of independent random variables with

the same law as TKi+1under PKi

ω [ · | TKi+1< TKi−1

]. Then, τ (j)− (i) is a sequence of

independent random variables with the same law as TKi (under PKi+1ω [ · | TKi < TKi+2

])and τ last,j

− is a sequence of independent random variables with the same law as TKi1−1

under PKi1ω [ · | TKi1−1

< Tnν ]. Clearly, the random variable τ(0) (respectively, τ(nν)) hasthe same law as TKi0+1

(respectively, Tnν ) under Pω[ · | TKi0+1< TKi0−1

] (respectively,

PKi1ω [ · | Tnν < TKi1−1

]).

In the reected case, we simply replace Ki by Ki, ξν(i) by ξνi and ω by ω.We want to give bounds on the number of backtracks between valleys before the

walk reaches bnνc. Denote

B(n) := cardi ≥ 1 : si+1(n) ≤ Tnν , Yi+1 < Yi =

i1−1∑i=1

ξν(i). (5.6)

175


K1 Ki0 0 Ki0+1 Ki−1 Ki Ki+1 Ki1 nν

time

Tinit

Tleft

next(i)

Tnext(i)

Tright

next∗(i1)

Fig. 6.3 On the decomposition (5.1) of Tnν

176

6. QUENCHED SLOWDOWN

By (2.8), we obtain that for i ≤ i1, P-a.s. for n large enough,

PKiω [TKi+1

> TKi−1] =

(Ki+1−1∑j=Ki−1

eV (j))−1

Ki+1−1∑j=Ki

eV (j) (5.7)

≤ maxi≤n

(Ki −Ki−1)(lnn)2/κ

n2/(1∧κ)

≤ n−3/2,

since maxi≤n(Ki+1 − Ki) ≤ (lnn)2 on A(n) and, due to Lemma 3.4, with the sameargument as for (3.10), we have V (Ki−1)−V (x) ≥ 2

1∧κ lnn− 2κ

ln lnn for x ∈ [Ki, Ki+1].Using (2.8) and (3.11), we obtain a lower bound : for ω ∈ A(n) ∩ F (n) ∩ G1(n) we

have

PKiω [TKi+1

> TKi−1] ≥ 1

Ki+1 −Ki−1

1

eV (Ki−1)−V (Ki+1)≥ n−(1+2γ0). (5.8)

During the rst 3n steps of the embedded random walk there are two cases, eitherthe walk has reached nν or there are at least n steps back. But then if nν is reachedin less than 3n steps, B(n) is stochastically dominated by a Bin(3n, n−3/2) by (5.7).Moreover, we get for f(·) such that f(n) = O(n), P-a.s. for n large enough,

Pω[B(n) ≥ f(n)] ≤(

3n

n

)( 1

n3/2

)n+ P

[Bin(3n, n−3/2) ≥ f(n)

],

and so using Stirling's formula and Chebyshev's exponential inequality, P-a.s. for n largeenough,

Pω[B(n) ≥ f(n)] ≤ exp(−C1n) + C2 exp(−f(n)) (5.9)

≤ C3 exp(−f(n)).

6 Quenched slowdown

In this section, we prove Theorem 1.2. Before going into technicalities, let us give aninformal argument about why we obtain dierent answers in Theorem 1.2.

Suppose that κκ+1

< 1 − νκ, or equivalently, ν < κ

κ+1. Consider the three strategies

depicted on Figure 6.4 :1 : The particle goes to the biggest valley in the interval [0, nν ], and stays there upto time n.

2 : The particle goes to the biggest valley in the interval [0, nκκ+1 ], stays there up to

time n− n κκ+1 , and then goes back to the interval [0, nν ].

3 : The particle goes to the biggest valley in the interval [−n κκ+1 , 0] (so that typically

it has to go roughly nκκ+1 units to the left), and stays there up to time n.

177


x

V (x)

0nν n

κκ+1

−nκ

κ+1

3

1

2

1κ+1 lnn

1κ+1 lnn

νκ lnn

time ≈ n

prob. ≈ exp(−nκ

κ+1 )

time ≈ n


κ+1 )

time ≈ nprob. ≈ exp(−n1− ν

κ )


κ+1 )

Fig. 6.4 The three strategies for the slowdown

178


By Lemmas 3.4 and 3.5, the biggest valley in the interval [0, nν ] has depth of approxi-mately ν

κlnn. Using Proposition 4.3, we obtain that the probability of staying there up

to time n is roughly exp(−n1− νκ ). As for the strategy 2, analogously we nd that the

biggest valley in the interval [0, nκκ+1 ] has depth around 1

κ+1lnn, and the probability

of staying there is roughly exp(−n κκ+1 ). Then, the probability of backtracking is again

around exp(−n κκ+1 ). The situation with the strategy 3 is the same as that with strat-

egy 2 (for the strategy 3, we rst have to backtrack and then to stay in the valley, butthe probabilities are roughly the same).

So, in the case ν < κκ+1

the strategies 2 and 3 are better than the strategy 1. Theonly situation when we cannot use neither 2 nor 3 is when the RWRE has reection inthe origin, and we are considering the hitting times.

6.1 Time spent in a valley

We have

Proposition 6.1. There exists γ4 > 0 such that for P-almost all ω, for all n largeenough we have for i ≤ 2n+ 1 and u ≥ 1,

PKiω

[TKi+1

> u(γ4(lnn)10eHi−1∨Hi

) | TKi+1< TKi−1

] ≤ e−u,

PKiω

[TKi−1


) | TKi−1< TKi+1

] ≤ e−u.

Démonstration. We prove only the second part of the proposition, the rst one uses thesame arguments. First, we have

maxx∈(Ki−1,Ki+1)

(max

y∈[x,Ki+1)V (y)− min

y∈[Ki−1,x)V (y)

)= Hi−1 ∨Hi.

Using (5.8) (or (5.7) for the rst part of the proposition), we obtain P-a.s. for n largeenough,

PKiω

[TKi−1


) | TKi+1> TKi−1

]≤ n1+2γ0PKi

ω

[TKi−1,Ki+1 > u

(γ4(lnn)10eHi−1∨Hi

), TKi+1

> TKi−1

].

To estimate this last probability, we may consider the random walk reected at Ki−1

and Ki+1. On A(n) we have Ki+1 − Ki−1 ≤ 2(lnn)2 and on G1(n) ∩ F (n) we havemaxy∈[Ki−1,Ki+1] V (y)−miny∈[Ki−1,Ki+1) V (y) ≤ 2γ0 lnn by (3.11). Hence for n such thatγ0 ≤ (lnn)2 we can apply Proposition 4.1 with a = Ki−1, c = Ki+1, M ≤ 2(lnn)2 andH = Hi−1 ∨Hi to get

PKiω

[TKi−1,Ki+1 > u

(γ4(lnn)10eHi−1∨Hi

)]≤ exp

(−uγ4(lnn)2/(32γ1)),

179


where ω denotes the environment with reection at Ki−1 and Ki+1, so that

PKiω

[TKi−1


) | TKi+1> TKi−1

]≤ exp

(−uγ4(lnn)2/(32γ1) + (1 + 2γ0) lnn)

≤ e−u,

for γ4 > 32γ1((1 + 2γ0) + 1) and n large enough.

Let Zi be a random variable with the same law as TKi+1under PKi

ω [ · | TKi+1< TKi−1

].Then, for i ∈ N(−na, nb) and H = maxi∈N(−na,nb) Hi, we have that P-a.s. for n largeenough

Ziγ4eH(lnn)10

≺ 1 + e, (6.1)

where e is an exponential random variable with parameter 1. Since ω ∈ G1(na∨b) P-a.s.for n large enough, there is a constant γ > 0 (depending only on κ) such that

Ziγ4n(a∨b)/κ(lnn)γ

≺ 1 + e. (6.2)

The same inequality is true when Ki−1 and Ki+1 are exchanged. We point out that the

same stochastic domination holds in the reected case, even for TKi0+2under P

Ki0+1

ω [ · |TKi0+2

< TKi0] = P

Ki0+1

ω [ · ] in which case it is a direct consequence of Proposition 4.1.Using the same kind of arguments as in the proof of Proposition 6.1 we obtain

Proposition 6.2. There exists a positive constant γ4 (without restriction of generality,the same as in Proposition 6.1) such that for P-almost all ω, we have for all n largeenough, with i0 = cardNn(−n, 0) and u ≥ 1,

Pω[TKi0+1(n) > u

(γ4(lnn)10eHi0−1∨Hi0

) | TKi0+1(n) < TKi0−1(n)

] ≤ e−u,

Pω[TKi0+1(n) > u


)] ≤ e−u.

Similarly we obtain

Proposition 6.3. There exists a positive constant γ4 (without restriction of generality,the same as in Proposition 6.1) such that for P-almost all ω, we have for all n largeenough with i1 = cardNn(−n, nν) and u ≥ 1,

PKi1ω

[Tnν > u


) | Tnν < TKi1 (n)

] ≤ e−u.

andPKi1ω

[TKi1−1

> u(γ4(lnn)10eHi1−1∨Hi1

) | TKi1−1< Tnν

] ≤ e−u.

This proposition implies that

τ last−γ4nν/κ(lnn)γ

≺ 1 + e. (6.3)

180


6.2 Time spent for backtracking

Recalling the denitions (5.5) and (5.6), we obtain, for the reected case,

Proposition 6.4. For 0 < a < b < c < 1, we have P-a.s. for n large enough,

Pω

[ Tbackγ4nν/κ(lnn)γ

≥ nc,B(n) ∈ [na, nb)]≤ exp(−nc/4),

where γ is as in (6.2).

Démonstration. On the event B(n) ∈ [na, nb), we have∑i∈N(0,nν) ξν(i) = B(n) < nb,

so we can use (6.2) and (6.3) to get that P-a.s. for n large enough,

Tbackγ4nν/κ(lnn)γ

≺ 2nb + Gamma(2nb, 1). (6.4)

(note that Tback is the time spent in valleys from 0 to nν because we have a reection at0). The factor 2 arises from the fact that each backtracking creates one right-left crossingand one left-right crossing. We use the following bound on the tail of Gamma(k, 1) :

P [Gamma(k, 1) ≥ u] ≤ e−u/2E[exp(Gamma(k, 1)/2)] = e−u/22k. (6.5)

Hence we have P-a.s. for n large enough,

Pω

[ Tbackγ4nν/κ(lnn)γ

≥ nc,B(n) ∈ [na, nb)]≤ P [Gamma(2nb, 1) ≥ nc − 2nb],

and since (nc− 2nb)/2− 2nb ln 2 ≥ nc/4 for n large enough, we conclude with (6.4).

In the same way, we get, still for the reected case

Proposition 6.5. For 0 < a < b < c < 1, we have P-a.s. for n large enough,

Pω

[ Tleftγ4nν/κ(lnn)γ

≥ nc,B(n) ∈ [na, nb)]≤ exp(−nc/4),

where γ only depends on κ.

Démonstration. On the event B(n) ∈ [na, nb), Tleft is lower than the time spent inthe valleys of indexes i0 and i0 + 1 during backtrackings from Ki0+1 to Ki0 . Since, thereare at most nb backtracks for this valley and since (6.2) is valid even for TKi0+2

under

PKi0+1

ω [ · ], we can use the same argument as in the proof of Proposition 6.4.

Next, recalling the denition (5.5), we obtain

181


Proposition 6.6. For 0 < a < b < 1 and c ∈ (b ∨ ν, 1), we have P-a.s. for n largeenough,

Pω

[ Tbackn(b∨ν)/κ(lnn)γ

≥ nc,B(n) ∈ [na, nb)]≤ exp(−nc/4),

where γ only depends on κ.

Démonstration. On the event B(n) ∈ [na, nb), Tback consists of the time spent in thevalleys indexed by Nn(−nb, nν), once this is noted we use the same argument as in theproof of Proposition 6.4.

6.3 Time spent for the direct crossing

We can control Tdir with the following proposition. Recall (3.7) and (3.8).

Proposition 6.7. For all m ≥ m0(κ, ν), we have for n large enough

Pω [Tdir ≥ n] ≤ C(m) exp(−n1−(1+2/m) νκ ).

Démonstration. Recall the denition (5.4) and let us take ω ∈ B′(n, ν,m) ∩G1(n). Letus introduce for k = −1, . . . ,m,

N(k) = cardi ∈ N(−nν , nν) : Hi ≥ νk

κmlnn+ 2 ln lnn, (6.6)

σ(k) = cardi ≤ Tnν : Xi ∈ [Kj(n), Kj+1(n)) for some j

with Hj ∈[lnn

νk

κm+ 2 ln lnn, lnn

ν(k + 1)

κm+ 2 ln lnn

]. (6.7)

If Tdir ≥ n, then for some k ∈ [−1,m] the particle spent an amount of time greaterthan n/(4m) in the valleys of depth in

[νkκm

lnn+ 2 ln lnn, ν(k+1)κm

lnn+ 2 ln lnn]because

ω is in G1(n), so that

Pω[Tdir > n] ≤ 4m maxk∈[−1,m]

Pω[σ(k) ≥ n/(4m)]. (6.8)

Using Proposition 6.1, since ω ∈ B′(n, ν,m)∩G1(n), we have N(k) ≤ nν(1−k/m), andso

σ(k)

γ4(lnn)11nν(k+1)/(κm)≺ 2nν(1−k/m) + Gamma(2nν(1−k/m), 1).

For m > (1 − ν)−1 we have that nν(1−k/m) = o(n1−ν(k+1)/m(lnn)−11), and for n largeenough (depending on ν and m), we use (6.5) to obtain

Pω[σ(k) ≥ n/(4m)] ≤ P[Gamma(2nν(1−k/m), 1) ≥ n1−ν(k+1)/(κm)

(lnn)12

]182


≤ 4nν(1−k/m)

exp(−n

1−ν(k+1)/(κm)

(lnn)12

)≤ exp

(−2n1−ν(k+2)/(κm) + ln 4nν(1−k/m)

).

We need to check that n1−(1+2/m)ν/κ ≥ ln 4nν(1−kε) for any k, if we take m largeenough, but this can be done by considering the cases k = 0 and k = m. Hence we getProposition 6.7.

6.4 Upper bound for the probability of quenched slowdown for

the hitting time

In this section we suppose that ω ∈ A(n)∩G1(n)∩B′(n, ν,m), which is satised P-a.s. for n large enough. First, we consider RWRE with reection at the origin. Becauseof (5.1)

Pω [Tnν > n] ≤Pω [Tdir ≥ n/5] + Pω [Tback ≥ n/5] + Pω [Tinit ≥ n/5] (6.9)

+ Pω [Tright ≥ n/5] + Pω [Tleft ≥ n/5] .

Let ε > 0 and recall (5.6), then

Pω [Tback ≥ n/5] ≤Pω[B(n) > n1−(1+2/m)ν/κ]

+ Pω[Tback ≥ n/5,B(n) ≤ n1−(1+2/m)ν/κ].

Using (5.9), we can write

Pω[B(n) > n1−(1+2/m)ν/κ] ≤ C2 exp(−n1−(1+2/m)ν/κ),

and for n large enough by Proposition 6.4,

Pω[Tback ≥ n/5,B(n) ≤ n1−(1+2/m)ν/κ]

≤ Pω

[ Tbacknν/κ(lnn)γ

≥ n1−(1+1/m)ν/κ,B(n) ≤ n1−(1+2/m)ν/κ]

≤ exp(−n1−(1+1/m)ν/κ/4)

≤ exp(−n1−(1+2/m)ν/κ),

so we obtainPω [Tback ≥ n/5] ≤ exp(−n1−(1+2/m)ν/κ). (6.10)

By Proposition 6.2, recalling (5.2), we have

Pω [Tinit ≥ n/5] ≤ exp(−n1−(1+2/m)ν/κ). (6.11)

183


Recalling 5.3, using Proposition 6.3 and the fact that ω ∈ G1(n), we get

Pω [Tright ≥ n/5] ≤ exp(−n1−(1+2/m)ν/κ). (6.12)

Finally, using (6.9), (6.10), (6.11), (6.12) and Proposition 6.7, we get that for allε > 0

Pω [Tnν > n] ≤ C3 exp(−n1−(1+2/m)ν/κ).

Hence, letting m go to ∞ we obtain

lim infn→∞

ln(− lnPω [Tnν > n])

lnn≥ 1− ν

κ, P-a.s. (6.13)

Now, we consider RWRE without reection. All estimates remain true except (6.10)for Tback. Concerning the estimates on Tleft it is easy to see that since Tleft > 0 impliesthat B(n) ≥ n/(lnn)2 − 1, we have using (5.9)

Pω[Tleft ≥ n/5] ≤ exp(−n1−(1+2/m)ν/κ). (6.14)

It remains to estimate Pω[Tback ≥ n], hence we take m and we note that

Pω[Tback > n] ≤m∑k=0

Pω[Tback > n,B(n) ∈ [nk/m, n(k+1)/m)].

Using (5.9), we obtain that P-a.s. for n large enough,

Pω[Tback > n,B(n) ∈ [nk/m, n(k+1)/m)] ≤ C3 exp(−nk/m).

Using Proposition 6.6, we obtain that

Pω[Tback > n,B(n) ∈ [nk/m, n(k+1)/m)] ≤ C4 exp(−C5n1−(ν∨((k+1)/m))/κ).

Hence, with these estimates on Tback, (6.9), (6.11), (6.14), (6.12) and Proposition 6.7 weobtain that P-a.s. for n large enough,

lim infn→∞


lnn≥ min

k∈[−1,m+1]

( km∨(

1− ν ∨ ((k + 1)/m)

κ

)),

minimizing we obtain,

lim infn→∞


lnn≥(

1− ν

κ

)∧ κ

κ+ 1− 2

(1 ∧ κ)m, P-a.s.

Taking the limit as m goes to innity yields the upper bound in (1.6), i.e.,

lim infn→∞


lnn≥(

1− ν

κ

)∧ κ

κ+ 1, P-a.s. (6.15)

184


6.5 Upper bound for the probability of quenched slowdown for

the walk

The argument of this section applies for both reected and non-reected RWREs,for the proof in the reected case, just replace Pω with Pω. We assume that ω ∈A(n) ∩G1(n) ∩B′(n, ν,m) which is satised P-a.s. for n large enough.

Set m ∈ Z+, we have using Markov's property

Pω[Xn < nν ] ≤m∑k=0

Pω[Tnν+(k−1)/m < n] (6.16)

×maxi≤n

P nν+(k−1)/m

ω [Xi < nν , Tnν+k/m > n− i].

First let us notice that

maxi≤n

P nν+(k−1)/m

ω [Xi ≤ nν , Tnν+k/m > n− i]

≤(

maxi≤n

P nν+(k−1)/m

ω [Xi < nν ])∧ P nν+(k−1)/m

ω [Tnν+k/m > n]. (6.17)

Using reversibility we have for any x ∈ Z (omitting integer parts for simplicity),

P nν+(k−1)/m

ω [Xi = x] ≤ π(x)

π(nν+(k−1)/m),

hence

maxi≤n

P nν+(k−1)/m

ω [Xi < nν ] ≤ 1 ∧ π([−n, nν ])π(nν+(k−1)/m)

.

Recall (2.3), then by (2.1) and the denition of bi we get π(bi1) ≤ 2e−V (bi1 ) and

π(bi1) ≤ 2e−V (bi1 ) ≤ C6(lnn)2/κn1/κe−V (Ki1+1(n)),

since, due to (3.8), the increase of potential in a valley is at most 1κ(lnn + 2 ln lnn).

Hence, using (3.10) and the fact that the width of the valleys is at most (lnn)2, we getthat

π([−n, nν ]) ≤ C7(lnn)2+2/κn1/κe−V (Ki1+1(n)).

Furthermore, denoting by i2 the index of the valley containing nν+(k−1)/m, for n largeenough we have using (2.1)

π(nν+(k−1)/m) ≥ π(Ki2−1(n)),

since on the eventG(n) both V (Ki2−1(n)) and V (Ki2−1(n)−1) are bigger than V (nν+(k−1)/m)and V (nν+(k−1)/m − 1).

185


On A(n), we have |(i2 − 1)− i1| ≥∣∣nν+(k−1)ε − nν∣∣ /(lnn)2 − 2. Since V (Ki) −

V (Ki+1) ≥ 1/(1 ∧ κ) lnn for ω ∈ G1(n), we have for k ≥ 2

π([−n, nν ])π(nν+(k−1)/m)

≤ C8(lnn)2+2/κn1/κ exp(−(V (Ki1+1)− V (Ki2−1))) (6.18)

≤ C9(lnn)2+2/κn1/κ exp(−C10

nν+(k−1)/m − nν(lnn)2

).

Moreover, using (1.6) in the non-reected case (or (6.13) in the reected case), wehave

P nν+(k−1)/m

ω [Tnν+k/m > n] ≤ exp(−n(1−(ν+(k/m))/κ)∧(κ/(κ+1))−1/m).

Hence, using this last inequality and (6.18), the inequality (6.16) becomes

Pω[Xn < nν ]

≤ maxk∈[−1,m+1]

[1 ∧

[C9mn

1/κ(lnn)2+2/κ exp(−C10

nν+(k−1)/m − nν(lnn)2

)]∧ exp(−n(1−(ν+(k/m))/κ)∧(κ/(κ+1))−1/m)

],

so that P-a.s.,

lim infn→∞


lnn≥ min

k∈[−1,m+1]

[(1k − 1

m≥ 0(ν +

k − 1

m

))∨((

1− ν + k/m

κ

)∧ κ

κ+ 1− 1

m

)].

Minimizing over k, we obtain

lim infn→∞


lnn≥(

1− ν

κ

)∧ κ

κ+ 1− 1

m, P-a.s.

Letting m goes to innity, we obtain

lim infn→∞


lnn≥(

1− ν

κ

)∧ κ

κ+ 1, P-a.s. (6.19)

6.6 Lower bound for quenched slowdown

In this section we assume ω ∈ A(n) ∩ D(n) ∩ F (n) which is satised P-a.s. for nlarge enough. First, we consider RWRE with reection at the origin.

For all ε > 0, note that for n large enough there is a valley of depth at least (1−ε)νκ

lnnstrictly before level nν and denote by i2 the index of the rst such valley. Hence

Pω[Tnν > n] ≥ PKi2ω [TKi2+1+1 > n],

186


and using Proposition 4.3 we obtain

PKi2ω [TKi2+1+1 > n] ≥ exp(−n1−(1−ε)ν/κ+ε).

Letting ε go to 0, yields

lim supn→∞


lnn≤ 1− ν

κ. (6.20)

This yields the lower bound for the exit time, so, recalling (6.13), we obtain (1.4).Now let us deduce the results on the slowdown. Set a ∈ [0, κ−ν), for n large enough

there is a valley of depth (ν+(1−ε)a)/κ lnn strictly before nν+a whose index is denotedi3. One possible strategy for the walk is to enter the i2-th valley at Ki2 + 1 ≤ nν+a, staythere up to time n−(nν+a−nν)−(lnn)2, then go to the left up to time n. The probabilityof this event can be bounded from below by

Pω[Xn < nν ] ≥Pω [Tnν+a < n/2] minj≤n

PeKi3+1

ω

[T eKi3−1, eKi3+1+1 > j

]× n−(3/ε0)(nν+a−nν+(lnn)2).

The rst term is bigger than 1/2 for n large enough (one can see this by using e.g. (6.20)).The second can be bounded by Proposition 4.3

minj≤n

PeKi3+1

ω

[T eKi3−1, eKi3+1+1 > j

]≥ exp(−n1−(ν+(1−ε)a)/κ+ε),

for n large enough. Then, the last term (going left) was dealt with using the fact thatω ∈ F (n).

This yields for any a ≥ 0,

lim supn→∞


lnn≤ 1a > 0(ν + a) ∨

(1− (1− ε)ν + a

κ+ ε),

and if we choose a = 0 ∨ (κ/(κ+ 1)− ν), we obtain

lim supn→∞


lnn≤(

1− ν

κ

)∧ κ

κ+ 1+

2ε

κ+ ε, P-a.s.

Together with (6.19), this yields (1.5) by letting ε go to 0.Now, we consider the case of RWRE without reection. Using the same reasoning,

we write

lim supn→∞


lnn≤ 1− ν

κ, P-a.s. (6.21)

187


Now we can see that, if we denote by i4 the index of a valley of depth at least(1 − ε)/(κ + 1) lnn between −nκ/(κ+1) and 0, since we are on D(n), we can go to thisvalley before reaching nν and then stay there for a time at least n. This yields,

Pω[Tnν > n] ≥ Pω[T−nκ/(κ+1) < Tnν ]PKi4ω [TKi4+1+1 > n],

bounding the rst term by the probability of going to the left on the nκ/(κ+1) rst steps,we get using Proposition 4.3 that for all n large enough

P 0ω [Tnν > n] ≥ n−(3/ε0)nκ/(κ+1)

exp(−n1−(1−2ε)/(κ+1)),

and hence

lim supn→∞

ln(− lnP 0ω [Tnν > n])

lnn≤ κ

κ+ 1+ 2

ε

κ+ 1, P-a.s. (6.22)

Moreover, it is clear thatPω[Xn < nν ] ≥ Pω[Tnν > n], (6.23)

and letting ε go to 0 in (6.22) and using (6.21) and (6.15), we obtain (1.6) and (1.7).This nishes the proof of Theorem 1.2.

7 Annealed slowdown

7.1 Lower bound for annealed slowdown

Let us dene the events

A′(n, ν, a) =there exists x ∈ [−nν , nν ] : max

y∈[x,nν ]V (y)− V (x) ≥ (1 + a) lnn

,

and

A′+(n, ν, a) =there exists x ∈ [0, nν ] : max

y∈[x,nν ]V (y)− V (x) ≥ (1 + a) lnn

.

Lemma 7.1. We have for a ∈ (−1, 1),

limn→∞

ln P[A′(n, ν, a)]

lnn= lim

n→∞

ln P[A′+(n, ν, a)]

lnn= −(κ− ν)− aκ.

Démonstration. From (2.7), it is straightforward to obtain that

P[A′+(n, ν, a)] ≤ P[A′(n, ν, a)]

≤ 2nνP[maxi≥0

V (i) ≥ (1 + a) lnn]

188

7. ANNEALED SLOWDOWN

= Θ(nν−(1+a)κ).

In order to give the corresponding lower bound, let us dene the event

A1(n, a) =there exists k ∈ [0, (lnn)2] such that V (k) ≥ (1 + a) lnn

,

we have

P[A1(n, a)] ≥P[maxi≥0

V (i) ≥ (1 + a) lnn]−P[V (lnn)2 > − lnn]

−P[

maxi≥(lnn)2

V (i)− V ((lnn)2) > (2 + a) lnn]

=Θ(n−(1+a)κ),

where we used (2.7) and a reasoning similar to the proof of Lemma 3.1. Now, we write

P[A′(n, ν, a)] ≥ P[A′+(n, ν, a)] ≥ nν

b(lnn)2cP[A1(n)] = Θ(nν−(1+a)κ

(lnn)2

),

and Lemma 7.1 follows.

For any ε > 0, on the event A′+(n, ν, ε) there exists a valley [Ki, Ki+1] with V (Ki+1)−V (bi) ≥ (1 + ε) lnn contained in [0, nν) and we denote by i5 its index. Then we have byProposition 4.2

Pω[Tnν > n] ≥ Pbi5ω [TKi5+1+1 > n] ≥ 1− γ2(1 + n)e−(1+ε) lnn ≥ 1

2

for n large enough. So

P[Tnν > n] ≥ E[1A′+(n, ν, ε)Pω[Tnν > n]] ≥ 1

2P[A′+(n, ν, ε)].

Hence we obtain by Lemma 7.1 that for any ε > 0

lim infn→∞

ln P[Tnν > n]

lnn≥ −(ν − κ)− κε.

Using (6.23), we obtain the corresponding lower bound for P[Xn < nν ] as well. ReplacingPω by Pω and P by P, exactly the same argument can be used to obtain the result inthe reected case.

189


7.2 Upper bound for annealed slowdown

We prove the upper bound in the non-reected case, the reected case follows easily ;indeed a simple coupling argument shows that Tnν in the environment ω is stochasticallydominated by Tnν in the environment ω. For m ∈ N such that 1/m ∈ (0, ν), we have

P[Tnν > n] ≤ P[A′(n, ν,−1/m)] + E(1A′(n, ν,−1/m)cP 0

ω [Tnν > n]).

The second term can be further bounded by

E(1A′(n, ν,−1/m)cP 0

ω [Tnν > n])

≤ P[A(n)c ∪B′(n, ν,m)c]

+ E(1A′(n, ν,−1/m)c ∩ A(n) ∩B′(n, ν,m)P 0

ω [Tnν > n]),

where B′(n, ν,m) is dened in (3.7).Using Lemma 7.1 we have that 1/n = o(P[A′(n, ν,−1/m)]), and thus Lemma 3.1

and Lemma 3.2 imply that

P[A(n)c ∪B′(n, ν,m)c] = o(P[A′(n, ν,−1/m)]).

We can turn (6.1) into the following, for i ∈ N(−nε, nν) we have

on A′(n, ν,−1/m)c ∩ A(n) ∩B′(n, ν,m),Z

C8n(1−1/m)(lnn)γ≺ 1 + e,

where Z has the same law as TKi+1(n) under PKi(n)ω [ · | TKi+1(n) < TKi−1(n)] ; γ = γ(κ)

and e denotes an exponential random variable of parameter 1. The same inequality istrue when Ki−1(n) and Ki+1(n) are exchanged.

This stochastic domination is the key argument for Section 6.4. We can adapt theproof of Proposition 6.4, so that on A′(n, ν,−1/m)c ∩ A(n) ∩ B′(n, ν,m) we obtain forall u ≥ 1,

Pω

[ Tbackn1−1/m(lnn)γ

≥ exp(n1/(2m)),B(n) ≤ n1/(4m)]≤ e−n

1/(2m)/4,

andPω

[Tright > n

5

]≤ C1 exp(−n1/(4m)).

Moreover, (5.9) still holds, so that

Pω[B(n) ≥ n1/(4m)] ≤ C2 exp(−n1/(4m)),

which yields

Pω

[Tleft > n

5

]≤ C3 exp(−n1/(4m)).

190

8. BACKTRACKING

Finally, recalling (5.2) and using Proposition 4.1 on A′(n, ν,−1/m)c∩A(n), we obtain

Pω

[Tinit > n

5

]≤ C4 exp(−n1/(4m)).

Since Proposition 6.7 remains true and A′(n, ν,−1/m)c ⊂ G(n), we get that for allω ∈ A′(n, ν,−1/m)c ∩ A(n) ∩B′(n, ν,m)

Pω[Tnν > n] ≤ C5 exp(−n1/(4m)).

Loosely speaking it costs at least exp(−n1/(2m)) to backtrack n1/m times, hence, onA′(n, ν,−1/m)c ∩ A(n) ∩ B′(n, ν,m), we can only see valleys of size lower than (1 −1/m) lnn. To spend a time n in those valleys would cost at least exp(−n1/(2m)). Thisnally implies that for all m > 0,

lim supn→∞

ln E[1A′(n, ν,−1/m)c, A(n)c, B′(n, ν,m)cP 0

ω [Tnν > n]]

lnn= 0,

so that

lim supn→∞

ln P[Tnν > n]

lnn≤ −(κ− ν) +

κ

m, (7.1)

the result for the hitting time follows by letting m go to innity.It is simple to extend this result to the position of the walk, indeed if Xn < nν then

Tn(1+1/m)ν > n or B(n) ≥ n1/(2m) and hence using (5.9) , we get for all m > 0

P[Xn < nν ] ≤ P[Tn(1+1/m)ν > n] + C6e−n1/(2m)

,

and the result follows by using (7.1) and letting m go to innity.This concludes the proof of Theorem 1.3.

8 Backtracking

In this section we prove Theorem 1.4.

8.1 Quenched backtracking for the hitting time

Set ν ∈ (0, 1) and consider Pω[T−nν < n]. First, we get that

for all ω ∈ F (n), Pω[T−nν < n] ≥ n−(3/ε0)nν ,

since the particle can go straight to the left during the rst nν steps, hence

lim supn→∞


lnn≤ ν. (8.1)

191


Secondly, we remark that if (−∞,−nν ] has been hit before time n then, at sometime i ≤ n the particle is at Xi ∈ [−n,−nν ] and hence for all ω

Pω[T−nν < n] ≤n∑i=1

Pω[Xi ∈ [−n,−nν ]]

≤ nmaxi≤n

Pω[Xi ∈ [−n,−nν ]]. (8.2)

In order to estimate this quantity, we use arguments similar to those in Section 6.5,i.e., rst we use the reversibility of the walk to write

maxi≤n

Pω[Xi ∈ [−n,−nν ]] ≤ π([−n,−nν ])π(0)

,

then, the right-hand side can be estimated in the same way as we obtained (6.18), andso we get on A(n) ∩G1(n) that

π([−n,−nν ])π(0)

≤ C1(lnn)2+2/κn1/κ exp(−C2nν/(lnn)2).

The previous inequality and (8.2) yield

for all ω ∈ A(n) ∩G(n), Pω[T−nν < n] ≤ C3n1+2/κ exp(−C2n

ν/(lnn)2),

so that

lim infn→∞


lnn≥ ν.

Together with (8.1), this proves (1.11).

8.2 Quenched backtracking for the position of the random walk

Let us denote a0 = κκ+1∨ ν. We give a lower bound for Pω[Xn < −nν ]. For n large

enough, there exists P-a.s. a valley of depth (1− ε)(a0/κ) lnn of index i2, between −na0

and 0. Consider the event that the walker goes to this valley directly and stays there upto time n − na0 and then goes to the left for the next na0 + 1 steps. On this event wehave Xn < −na0 , so we obtain

Pω[Xn < −nν ] ≥ n−(3/ε0)2(na0+1)PKi2+1−1ω [TKi2−1,Ki2+1+1 ≥ n]

≥ n−(3/ε0)2(na0+1) exp(−n1−(1−2ε)a0/κ),

where we used Proposition 4.3 and ω ∈ F (n). Hence we obtain

lim supn→∞


≤ a0 +2εa0

κ,

192

8. BACKTRACKING

and letting ε go to 0 we have

lim supn→∞


≤ a0. (8.3)

Turning to the upper bound, we have for m ∈ N,

Pω[Xn < −nν ] ≤m∑k=0

Pω[Tn(k−1)/m < n] maxi≤n

P n(k−1)/m

ω [Tnk/m > n− i,Xi < −nν ], (8.4)

where once again

maxi≤n

P n(k−1)/m

ω [Tnk/m > n− i,Xi < −nν ]

≤(

maxi≤n

P n(k−1)/m

ω [Xi < −nν ])∧ P n(k−1)/m

ω [Tnk/m > n].

First, using (1.6), for n large enough

P n(k−1)/m

ω [Tnk/m > n] ≤ exp(−n(1−(k/m)/κ)∧(κ/(κ+1))−1/m). (8.5)

Then, as in Section 6.5, the reversibility of the walk yields that

maxi≤n

P n(k−1)/m

ω [Xi ∈ [−n,−nν ]] ≤ π([−n,−nν ])π(n(k−1)/m)

, (8.6)

the right-hand side can be estimated in the same way we obtained (6.18) and we get onA(n) ∩G(n)

π([−n,−nν ])π(n(k−1)/m)

≤ C4 exp(−C5(n(k−1)/m + nν)/(lnn)2). (8.7)

Putting together (8.4), (8.5), (8.6), and (8.7), we obtain

lim infn→∞


≥ mink∈[0,m]

(((1− k

mκ

)∧ κ

κ+ 1

)∨(k − 1

m∨ ν))− 1

m,

minimizing yields that

lim infn→∞


≥ a0 − 2

m,

letting m go to innity and recalling (8.3) we obtain (1.9).

193


8.3 Annealed backtracking

Let θ0 = E [ln ρ0] < 0. Dene

R =ω : V (x) ≤ θ0

3nν for x ∈ [0, n], |V (x) + θ0x| ≤ |θ0|

3nν for x ∈ [−nν , 0)

.

Since V is a sum of i.i.d. random variables having some nite exponential moments, wecan use large deviations techniques to obtain C6 such that

P[R] ≥ 1− 2ne−C6nν . (8.8)

Then, on R, using (2.8), we obtain

Pω[T−nν < n] ≤ Pω[T−nν < Tn]

≤ C7n exp(−2θ0

3nν). (8.9)

Using (8.8) and (8.9), we obtain

P[Xn < −nν ] ≤ P[T−nν < n] ≤ e−C8nν . (8.10)

On the other hand, we easily obtain that

P[T−nν < n] ≥ P[Xn < −nν ] ≥(δ

2

)nνn−C9 , (8.11)

where δ > 0 is such that P[1 − ω0 ≥ δ] > 1/2. Indeed on the event of probability atleast (1/2)n

νthat 1− ωx ≥ δ for x ∈ (−nν , 0], the particle can go directly (to the left

on each step) to (−nν), and then the cost of creating a valley of depth 2 lnn there ispolynomial and then it costs nothing to stay there for a time n by Proposition 4.2. Now,(8.10) and (8.11) imply (1.10). This nishes the proof of Theorem 1.4.

9 Speedup

In this section we prove Theorem 1.5. So, we have κ < 1, ν ∈ (κ, 1) ; let us denoteg(α) = ν + α

κ− α, and let α0 = κ 1−ν

1−κ . Clearly, g(α) is a linear function, g(0) = ν < 1,g(ν) = ν

κ> 1, and g(α0) = 1 ; note also that ν − α0 = ν−κ

1−κ .The discussion in this section is for the RWRE on Z (i.e., without reection), the

proof for the reected case is quite analogous.

194

9. SPEEDUP

9.1 Lower bound for the quenched probability of speedup

We are going to obtain a lower bound for Pω[Xn > nν ].By Lemma 3.2 and Borel-Cantelli, for any xedm, ω ∈ B′(n, α0,m)∩A(n)∩F (n) for

all n large enough, P-a.s. (recall the denition of A(n) and B′(n, α0,m) from Section 3).So, from now on we suppose that ω ∈ B′(n, α0,m) ∩ A(n).

Let us denote M = Nn(0, nν), dene the index sets

I0 = i ∈M : Hi−1 ∨Hi ≤ ln lnn,Ik =

i ∈M : (Hi−1 ∨Hi)− ln lnn ∈

[(k − 1)α0

mκlnn,

kα0

mκlnn)

for k ∈ [1,m− 1], and

U =i ∈M : Hi−1 ∨Hi ≥ (m− 1)α0

mκlnn+ ln lnn

.

Note that on B′(n, α0,m)

cardU ≤ nν−α0+α0m = n

ν−κ1−κ+

α0m , (9.1)

card Ik ≤ nν−kα0m , for all k = 1, . . . ,m− 1. (9.2)

Recalling (2.3) we dene the quantities σi0 = TKi0+1, σi1 = Tnν − TKi1 , and σj =

TKj+1− TKj for j = i0 + 1, . . . , i1 − 1. Then for ε > 0, we can write

Pω[Xn > n(1−ε)ν ] ≥Pω[m−1∑k=0

∑i∈Ik

σi ≤ n

2

]Pω

[∑i∈U

σi ≤ n

2

]× P nν

ω

[Xj > n(1−ε)ν for all i ∈ [0, n− nν ]]. (9.3)

Let us obtain lower bounds for the three terms in the right-hand side of (9.3). First,we write using (9.2)

Pω

[m−1∑k=0

∑i∈Ik

σi ≤ n

2

]≥

m−1∏k=0

Pω

[∑i∈Ik

σi ≤ n

2m

]≥

m−1∏k=0

Pω

[σi ≤ 1

2mn1−(ν− kα0

m) for all i ∈ Ik

]. (9.4)

Now, consider any ` ∈ Ik and write

Pω

[σ` ≤ 1

2mn1−(ν− kα0

m)]

195


≥ PK`ω [TK`+1

< TK`−1]

× PK`ω

[TK`−1,K`+1 ≤

1

2mn1−(ν− kα0

m) | TKl+1

< TKl−1

].

By the formula (5.7), on A(n) we have

PK`ω [TK`+1

< TK`−1] ≥ 1− n−3/2,

and by Proposition 6.1,

PK`ω

[TK`−1,K`+1 ≤

1

2mn1−(ν− kα0

m) | TKl+1

< TKl−1

]≥ 1− exp

(− C1

m(lnn)γn1−(ν− kα0

m)− kα0

mκ

),

so

Pω

[σ` ≤ 1

2mn1−(ν− kα0

m)]≥ (1− n−3/2)

(1− exp

(− C1

m(lnn)γn1−g( kα0

m))). (9.5)

Now, for k ≤ m− 1 we have

1− g(kα0

m

)≥ (1− κ)α0

mκ,

so (9.4) and (9.5) imply that

Pω

[m−1∑k=0

∑i∈Ik

σi ≤ n

2

]≥

m−1∏k=0

[(1− n−3/2)

)(1− exp

(− C1

m(lnn)γn

(1−κ)α0mκ

))]nν→ 1 as n→∞. (9.6)

Now, we obtain a lower bound for the second term in the right-hand side of (9.3).On G1(n), we get an upper bound on ρi for i ∈ [−n, n] and hence we have ωx ≥ n−C2 ,we obtain for any ` ∈ U (imagine that, to cross the corresponding interval, the particlejust goes to the right at each step)

Pω

[σ` ≤ 1

2n1−(ν−α0)−α0

m

]≥ n−C2(lnn)2 , (9.7)

so,

Pω

[∑i∈U

σi ≤ n

2

]≥ Pω

[σ` ≤ 1

2n1−(ν−α0)−α0

m for all ` ∈ U]

≥ (n−C2(lnn)2)n ν−κ1−κ+

α0m

196

9. SPEEDUP

= exp(−C2(lnn)3n

ν−κ1−κ+

α0m

)(9.8)

(recall that ν − α0 = ν−κ1−κ ).

As for the third term in (9.3), using (2.8) we easily obtain that, on A(n) ∩G(n),

P nν

ω

[Xj > n(1−ε)ν for all j ∈ [0, n− nν ]] ≥ P nν

ω [Tn < Tn(1−ε)ν ] > C3 > 0. (9.9)

Now, plugging (9.6), (9.8), and (9.9) into (9.3) and sending m to ∞, we obtain that

lim supn→∞

ln(− lnPω[Xn > n(1−ε)ν ])

lnn≤ ν − κ

1− κ , P-a.s.

applying this for ν ′ = ν/(1− ε) and letting ε go to 0,

lim supn→∞

ln(− lnPω[Xn > nν′])

lnn≤ ν ′ − κ

1− κ , P-a.s. (9.10)

Since obviously Pω[Tnν < n] ≥ Pω[Xn > nν ], (9.10) holds for Pω[Tnν < n] as well.

9.2 Upper bound for the quenched probability of speedup

Fix ε > 0 such that α0 + ε < ν. Dene

W =i ∈ Nn(0, nν) : Hi ≥ α0 + ε

κlnn− 4 ln lnn

,

Ψεn =

ω : cardW ≥ 1

3nν−α0−ε

.

By Lemma 3.5, on each subinterval of length nα0+ε we nd a valley of depth at leastα0+εκ

lnn−4 ln lnn with probability at least 1/2. Since the interval [0, nν ] contains nν−α0−ε

such subintervals, we have

P[Ψεn] ≥ 1− exp(−C4n

ν−α0−ε), (9.11)

in particular by Borel-Cantelli's Lemma, P-a.s. we have ω ∈ Ψεn for n large enough.

For i ∈ W , dene σi = TKi+1+1 − TKi+1, and let

s0 =1

4γ2(lnn)4nα0+εκ .

Then, by Proposition 4.2, for any i ∈ W ,

Pω[σi < s0] ≤ 2γ2s0 exp(−α0 + ε

κlnn+ 4 ln lnn

)= 2γ2s0n

−α0+εκ (lnn)4

197


=1

2. (9.12)

Dene the family of random variables ζi = 1σi < s0, i ∈ W . These random vari-ables are independent with respect to Pω, and Pω[ζi = 1] ≤ 1/2 by (9.12). Supposewithout restriction of generality that (recall that g(α0) = 1)

1

3s0 × 1

3nν−α0−ε =

1

36γ2(lnn)4ng(α0+ε) > n.

Then, since cardW ≥ 13nν−α0−ε for ω ∈ Ψε

n, we see using large deviations techniquesthat for n large enough

Pω[Tnν < n] ≤ Pω

[∑i∈W

ζi >2

3cardW

]≤ exp

(−C5nν−κ1−κ−ε

)(9.13)

(recall that ν − α0 = ν−κ1−κ ). Since ε > 0 is arbitrary, we obtain

lim infn→∞

ln(− lnPω[Tnν < n])

lnn≥ ν − κ

1− κ P-a.s. (9.14)

Together with (9.10), this shows (1.12).

9.3 Annealed speedup

As usual, the quenched lower bound obtained in Section 9.1 also yields the annealedone, i.e. (9.10) implies that

lim supn→∞

ln(− ln P[Xn > nν ])

lnn≤ ν − κ

1− κ , (9.15)

Turning to the upper bound, we have by (9.11) and (9.13) that

P[Tnν < n] =

∫Pω[Tnν < n] dP

≤∫

Ψεn

Pω[Tnν < n] dP + P[(Ψεn)c]

≤ exp(−C5n

ν−κ1−κ−ε

)+ exp

(−C4nν−κ1−κ−ε

),

and this implies (1.13). This nishes the proof of Theorem 1.5.

198

9. SPEEDUP

Acknowledgements

A.F. would like to thank the ANR MEMEMO, the Accord France-Brésiland theARCUS program.

S.P. is thankful to FAPESP (04/072762), CNPq (300328/20052 and 471925/20063), and N.G. and S. P. are thankful to CAPES/DAAD (Probral) for nancial support.

We thank two anonymous referees whose extremely careful lecture of the rst versionlead to many improvements.

199


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