Extreme statistics of random walks : a real space ...Extreme statistics of random walks : a real space renormalization group approach G. Schehr Laboratoire de Physique Théorique Orsay,

Extreme statistics of random walks : a real spacerenormalization group approach

G. Schehr

Laboratoire de Physique ThéoriqueOrsay, Université Paris XI

G. S., P. Le Doussal, J. Stat. Mech.-Theory E., P01009 (2010), arXiv:0910.4913

G.Schehr (LPT Orsay) EVS of random walks : a RSRG approach DDAP 6, Sydney 1 / 14

Purpose

xm

T

x(t)

timetm

xm ≡ max0≤t≤T

x(t)

x(tm) = xm

x(t) ≡

Brownian motion (BM)Continous Time Random WalkBessel process (radius of d-dimensional BM)


Purpose

xm

T

x(t)

timetm

xm ≡ max0≤t≤T

x(t)

x(tm) = xm

xm, tm ≡ random variables


Purpose

xm

T

x(t)

timetm

xm ≡ max0≤t≤T

x(t)

x(tm) = xm

xm, tm ≡ random variables

What are the probability distribution functions of xm, tm ?


Motivations

xm, tm are fundamental characteristics of stochastic processesLévy 1939, Sparre-Andersen 1954

For free BM: xmd= ’local time’ , tm

d= ’occupation time’

e.g. , PT (tm) =1TP̃�

tmT

�, P̃(z)

1π�

z(1 − z)

Related to hitting probability Majumdar, Rosso, Zoia 2010Applications to the convex hull of 2d Brownian motion

Randon-Furling, Majumdar, Comtet 2009

Extreme value statisticsin engineeringin financestatistical physics of disordered systems:e.g. diffusion in random environment


Motivations

xm, tm are fundamental characteristics of stochastic processesLévy 1939, Sparre-Andersen 1954

For free BM: xmd= ’local time’ , tm

d= ’occupation time’

e.g. , PT (tm) =1TP̃�

tmT

�, P̃(z)

1π�

z(1 − z)

Related to hitting probability Majumdar, Rosso, Zoia 2010Applications to the convex hull of 2d Brownian motion

Randon-Furling, Majumdar, Comtet 2009

Extreme value statisticsin engineeringin financestatistical physics of disordered systems:e.g. diffusion in random environment


Extreme values statistics (EVS) : what do we know ?

identical & independent random variables

x(1), x(2), ..., x(N) ≡ N rand. var., P0(x)

xm = max1≤i≤N x(i) , Q : PN(xm) for N � 1 ?

Three universality classes depending on P0(x → ∞)

for random walks, x(t), t ∈ [0,T ] are strongly correlated

e.g. for BM, �x(t1)x(t2)� − �x(t1)��x(t2)� = min(t1, t2)

EVS of strongly correlated variables : a hard problem !


EVS of random walks

Markov random walks : a paradigm for EVS of strongly correlatedvariables

Analytical tools :

(Backward) Fokker-Planck equation

Feynman-Kac formula (path integrals methods)

Real space renormalization method (HERE)• Ma and Dasgupta (1980):

disordered quantum spin chains• D.S. Fisher (1994)• Le Doussal and Monthus (2003):

particle in a random environment


Real space renormalization group : first step

x(0) = 0 , x(i) = x(i − 1) + η(i)

Identify the local extrema

x(0)

x(i)

0 i

ti

x̃(ti)

time

time

T


Real space renormalization group : second step

Decimation of the smallest ”jump”

F3

F2

F1

�1 �2 �3

F = F1 − F2 + F3


Real space renormalization at work











tm

xm


Real space renormalization : the hard part

Translate this decimation procedure into equations→ RG equations for the pdf of renormalized paths

For Markov processes : RG equations for joint pdf PΓ(F , �)

F3

F2

F1

�1 �2 �3

F = F1 − F2 + F3

� = �1 + �2 + �3

Solve these equations and find fixed pointsObtain an analytic expression for the joint pdf PT (tm, xm)


Results for Brownian Motion

x(0) = 0 , x(i) = x(i − 1) + η(i) ,η(i) are i.i.d such �η(i)2� < ∞

Distribution of the maximum xm

PT (xm) =1

T 12

P̃�

xm

T 12

�, P̃(z) = θ(z)

e− z24

√π

Distribution of the position of the maximum tm

PT (tm) =1TP̃�

tmT

�, P̃(z) =

1π�

z(1 − z)


Results for Continous Time Random Walks

A model for anomalous diffusion Montroll, Weiss 1965

x(0) = 0 , x(i) = x(i − 1) + η(i) ,η(i) are i.i.d such �η(i)2� < ∞

+ waiting time τ between each jump with a broad distributionψ(τ) ∼ τ−1−α, α < 1

Subdiffusive behavior

Typical number of steps n ∼ tα

Typical displacement x(t) ∼ n1/2 ∼ tα/2



Distribution of the maximum xm G.S., Le Doussal 2010

PT (xm) =1

T α2

P̃� xm

T α2

�, P̃(z) = θ(z)

2α

z−(1+ 2α )Lα

2

�z− 2

α

�

Lν(z) ≡ one sided stable distribution ,

� ∞

0Lν(z) e−pzdz = e−pν

Lν(z) =1π

∞�

n=1

(−1)n+1z−νn−1 Γ(nν + 1)n!

sin (nπν)

∼�

exp�−κz− ν

1−ν�, z → 0

z−(1+ν) , z → ∞



Distribution of the maximum xm G.S., Le Doussal 2010

PT (xm) =1

T α2

P̃� xm

T α2

�, P̃(z) = θ(z)

2α

z−(1+ 2α )Lα

2

�z− 2

α

�

∼ exp(−κ z2

2−α)

z

P̃(z)

∼ 1Γ(1−α/2) − a z



Distribution of the position of the maximum tm G.S., Le Doussal 2010

PT (tm) =1TP̃�

tmT

�, P̃(z) =

sin�απ2

�

π

1z1−α

2 (1 − z)α2



Distribution of the position of the maximum tm G.S., Le Doussal 2010

PT (tm) =1TP̃�

tmT

�, P̃(z) =

sin�απ2

�

π

1z1−α

2 (1 − z)α2

1

2

3

4

0 0.2 0.4 0.6 0.8 1z

P̃(z)

α = 1/3

α = 1


Conclusions and perspectives

Real space renormalization : a powerful method to obtainanalytical results for the extreme statistics of random walks

New exact results for the extreme statistics CTRW and Besselprocesses

Extension to other stochastic processes ?

Applications to diffusion in disordered environment ?


Documents

Extreme statistics of random walks : a real space ...Extreme statistics of random walks : a real space renormalization group approach G. Schehr Laboratoire de Physique Théorique Orsay,