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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)
G.Schehr (LPT Orsay) EVS of random walks : a RSRG approach DDAP 6, Sydney 2 / 14
Purpose
xm
T
x(t)
timetm
xm ≡ max0≤t≤T
x(t)
x(tm) = xm
xm, tm ≡ random variables
G.Schehr (LPT Orsay) EVS of random walks : a RSRG approach DDAP 6, Sydney 2 / 14
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 ?
G.Schehr (LPT Orsay) EVS of random walks : a RSRG approach DDAP 6, Sydney 2 / 14
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
G.Schehr (LPT Orsay) EVS of random walks : a RSRG approach DDAP 6, Sydney 3 / 14
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
G.Schehr (LPT Orsay) EVS of random walks : a RSRG approach DDAP 6, Sydney 3 / 14
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 !
G.Schehr (LPT Orsay) EVS of random walks : a RSRG approach DDAP 6, Sydney 4 / 14
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
G.Schehr (LPT Orsay) EVS of random walks : a RSRG approach DDAP 6, Sydney 5 / 14
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
G.Schehr (LPT Orsay) EVS of random walks : a RSRG approach DDAP 6, Sydney 6 / 14
Real space renormalization group : second step
Decimation of the smallest ”jump”
F3
F2
F1
�1 �2 �3
F = F1 − F2 + F3
G.Schehr (LPT Orsay) EVS of random walks : a RSRG approach DDAP 6, Sydney 7 / 14
Real space renormalization at work
G.Schehr (LPT Orsay) EVS of random walks : a RSRG approach DDAP 6, Sydney 8 / 14
Real space renormalization at work
G.Schehr (LPT Orsay) EVS of random walks : a RSRG approach DDAP 6, Sydney 8 / 14
Real space renormalization at work
G.Schehr (LPT Orsay) EVS of random walks : a RSRG approach DDAP 6, Sydney 8 / 14
Real space renormalization at work
G.Schehr (LPT Orsay) EVS of random walks : a RSRG approach DDAP 6, Sydney 8 / 14
Real space renormalization at work
G.Schehr (LPT Orsay) EVS of random walks : a RSRG approach DDAP 6, Sydney 8 / 14
Real space renormalization at work
tm
xm
G.Schehr (LPT Orsay) EVS of random walks : a RSRG approach DDAP 6, Sydney 8 / 14
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)
G.Schehr (LPT Orsay) EVS of random walks : a RSRG approach DDAP 6, Sydney 9 / 14
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)
G.Schehr (LPT Orsay) EVS of random walks : a RSRG approach DDAP 6, Sydney 10 / 14
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
G.Schehr (LPT Orsay) EVS of random walks : a RSRG approach DDAP 6, Sydney 11 / 14
Results for Continous Time Random Walks
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 → ∞
G.Schehr (LPT Orsay) EVS of random walks : a RSRG approach DDAP 6, Sydney 12 / 14
Results for Continous Time Random Walks
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
G.Schehr (LPT Orsay) EVS of random walks : a RSRG approach DDAP 6, Sydney 12 / 14
Results for Continous Time Random Walks
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
G.Schehr (LPT Orsay) EVS of random walks : a RSRG approach DDAP 6, Sydney 13 / 14
Results for Continous Time Random Walks
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
G.Schehr (LPT Orsay) EVS of random walks : a RSRG approach DDAP 6, Sydney 13 / 14
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 ?
G.Schehr (LPT Orsay) EVS of random walks : a RSRG approach DDAP 6, Sydney 14 / 14
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