Random walks in Beta random environment
Guillaume Barraquand
22 mai 2015
(Based on joint works with Ivan Corwin)
Consider the simple random walk Xt on Z, starting from 0. We note
¡
¢
P Xt+1 = Xt + 1 =
α
α+β
¡
¢
, P Xt+1 = Xt − 1 =
The Central Limit Theorem says that
α−β
Xt − t α+β
=⇒ N (0, 1).
p
σ t
Theorem (Cramér)
For
α−β
α+β
< x < 1,
³
´
log P(Xt > xt)
t
−−−→ −I(x),
t→∞
where I(x) is the Legendre transform of
¶
µ z
³ £
¤´
αe + βe−z
.
λ(z) := log E ezX1 = log
α+β
β
α+β
.
In random environment ?
Question
What can we say for a random walk in random environment ?
In this talk, we consider simple random walks on Z in space-time i.i.d.
environment:
P Xt+1 = x + 1|Xt = x
¡
¢
= Bt,x ,
P Xt+1 = x − 1|Xt = x
¡
¢
= 1 − Bt,x ,
where (Bt,x )t,x are i.i.d.
We note P, E (resp. P, E) the measure and expectation with respect to
the random walk (resp. the environment)
Answer
All results from the previous slide still hold, even conditionally on the
environment, for almost every realization of the environment.
Quenched large deviation principle
Theorem (Rassoul-Agha, Seppäläinen and Yilmaz, 2013)
Assume that log(Bt,x ) have a finite third moment. Then, the limiting
moment generating function
³ £
¤´
1
log E ezXt ,
t→∞ t
λ(z) := lim
exists a.s., and
log
P(Xt > xt)
³
´
t
a. s .
−−−→ −I(x).
t→∞
where I(x) is the Legendre transform of λ.
An exactly solvable model: the Beta RWRE
We assume that (Bt,x ) follow the Beta(α, β) distribution.
¡
¢
Γ(α + β)
P B ∈ [x, x + dx] = xα−1 (1 − x)β−1
dx.
Γ(α)Γ(β)
Ï Exactly solvable means
that we can exactly
compute the law of
P(Xt > xt) (and more).
Ï The annealed law of the
Beta RWRE is the
simple random walk
from the first slide.
Xt
t
0
x
Bx , t
(x, t)
1 − Bx,t
For simplicity, assume α = β = 1. (Uniform case)
Theorem (B.-Corwin)
The LDP rate function is
I(x) = 1 −
p
1 − x2 .
We have the convergence in distribution as t → ∞,
³ ¡
¢´
log P Xt > xt + I(x)t
=⇒ L GUE ,
σ(x) · t1/3
where L GUE is the GUE Tracy-Widom distribution, and
σ(x)3 =
2I(x)2
,
1 − I(x)
under the (technical) hypothesis that x > 4/5.
The theorem should extend to the general parameter case α, β and
when x covers the full range of large deviation events (i.e. x ∈ (0, 1)).
Fredholm determinant
Theorem (B.- Corwin)
Let u ∈ C \ R+ , and t, x with the same parity. Then for any parameters
α, β > 0 one has
i
h
E euP(Xt >x) = det(I + Ku )L2 (C0 )
where C0 is a small positively oriented circle containing 0 but not −α − β
nor −1, and Ku : L2 (C0 ) → L2 (C0 ) is defined by its integral kernel
1
2iπ
Ku (w, w0 ) =
Z
1/2+i∞
1/2−i∞
π
sin(πs)
(−u)s
g(w)
ds
g(w + s) s + w − w0
where
Γ(w)
g(w) =
Γ(α + w)
µ
¶(t−x)/2 µ
Γ(α + β + w)
Γ(α + w)
¶(t+x)/2
Γ(w).
Z
h
in
∞ 1 µ 1 ¶n Z
X
...
det Ku (wi , wj )
dw1 . . . dwn .
i,j=1
C0
C0
n=1 n! 2iπ
det(I+Ku )L2 (C0 ) := 1+
Idea of the proof
Direct proof
1. Interpret the r.v. P(Xt > x) as the partition function Z(t, x) of some
polymer model (a particular random average process).
2. Find a recurrence relation for Z(t, x).
3. It yields an evolution equation for t 7→ E [Z(t, x1 ) . . . Z(t, xk )].
4. Solve the equation using a variant of Bethe ansatz.
5. Take moment generating series. It works !
6. Write it as a Fredholm determinant using ideas from Macdonald
processes.
Origin
Z(t, x) is a limit of observables of the q-Hahn TASEP, a Bethe ansatz
solvable interacting particle system introduced by Povolotsky. (like the
strict-weak polymer, cf Hao Shen’s talk)
Extreme value theory
Fact
The order of the maximum of N i.i.d. random variables is the quantile
or order 1 − 1/N.
Relation LDP / extreme values
Second order corrections to the LDP have an interpretation in terms of
second order fluctuations of the maximum of i.i.d. drawings.
Corollary (B.-Corwin)
Let Xt(1) , . . . , Xt(N) be random walks drawn independently in the same
environment. Set N = ect . Then, for α = β = 1,
n
o p
maxi=1,...,ect Xt(i) − t 1 − (1 − c)2
=⇒ L GUE ,
d(c) · t1/3
where d(c) is an explicit function (proved under assumption c > 2/5).
Zero temperature limit
We define the first passage-time T(n, m)
from (0, 0) to the half-line Dn,m by
X
T(n, m) =
min
te
π:(0,0)→Dn,m e∈π
Dn,m
Passage times
m
(0, 0)
n
For (ξi,j ) i.i.d. Bernoulli and (Ee ) i.i.d.
Exponential,
(
ξi,j Ee
if e is horizontal,
te =
(1 − ξi,j )Ee if e is vertical.
Theorem (B.-Corwin)
For any κ > a/b and parameters a, b > 0, there exist constants ρ (κ) and
τ(κ), s.t.
T(n, κn) − τ(κ)n
=⇒ L GUE .
ρ (κ)n1/3
Dynamical construction
Alternative description
Ï At time 0, only one random
walk trajectory (in black).
Ï One adds to the percolation
cluster portions of
branching-coalescing
random walks at
exponential rate, at each
branching point.
Outlook
We have seen
Ï A first exactly solvable model of space-time RWRE.
Ï Second order corrections to the LDP converge to L GUE .
Ï Limit theorem for the max of N = ect trajectories.
Ï Results propagate to the zero temperature model.
Questions
Ï KPZ universality for RWRE and random average process, to which
extent ?
Ï Integrability : determinantal structure ? Analogue of
Schur/Macdonald processes ? Link with a random matrix model ?
Ï Tracy-Widom distribution and extreme value theory...