Introduction to random walks in random and non-random environments

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Introduction to random walks in random and non-random environments

Nadine Guillotin-Plantard

Institut Camille Jordan - University Lyon I

Grenoble – November 2012

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Outline

1 Simple Random Walks in Z^d Definition

Recurrence - Transience

Asymptotic distribution forn large Asymmetric random walk

2 Random Walks in Random Environments Definition

Recurrence-Transience

Valleys (or traps) - Slowing down Asymptotic distributions forn large

3 Random Walk in Random Scenery

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Simple Random Walks inZ^d Definition

At time 0, a walker starts from the site 0, tosses a coin. If he gets ”Head”, then he goes to the site +1, otherwise to the site -1. (”Tail”)

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Natural questions

Does the walker come back to the origin ? Notion of Recurrence - Transience.

Mean position of the walker, fluctuations around this position, large deviations...

Probability that the walker be at site x at timen (Local limit theorem)

Number of distinct sites visited by the walker up to timen (Range) Maximal (or minimal) position of the walker before time n

Number of visits to a fixed sitex. (Local time ) The last time the random walker visits 0 before time n

The number of positive values of the random walk before time n Number of self-intersections up to time n.

Favorite sites of the walker and so on...

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Let (X_i)_i≥1 be i.i.d. random variables taking values +1 or−1 with equal probability.

{X_i = +1}={ The walker gets ”Head” at timei}.

The position of the walker at time n is given by : S₀:= 0

and for any n≥1,

S_n:=

n

X

i=1

X_i (S_n)n≥0 is called simple random walk onZ. From this writing, we can compute

E(S_n) = 0 and Var(Sn) =E(S_n²) =n. Therefore,Sn∼√

n

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Simple Random Walks inZ^d Recurrence - Transience

For n integer,

P(S_2n= 0) = Number of paths of length 2n from 0 to 0 Number of paths of length 2n

= C_2nⁿ 2²ⁿ

= (2n)!

4ⁿ(n!)²

∼ 1

√πn for n large using Stirling’s formula

n!∼n e

n√ 2πn.

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A random walk is said recurrent iff Ph

lim sup

n

{S_n= 0}i

=Ph

S_n= 0 i.o.i

= 1

Otherwise, it is called transient. SinceS_n is a Markov chain, we have this useful criterion :

Theorem

(S_n)_n is recurrent iff

+∞

X

n=0

P(S_n= 0) = +∞

Since P(S_n= 0)∼C/√

n, the simple random walk onZ isrecurrent.

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Simple random walk in Z

²

Simple random walk in Z²

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Georges P´olya (1887 – 1985)

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In higher dimension

Theorem (P´olya (1921) )

There exists some constant C =C(d) s.t. for n large enough P(S_n= 0)∼C n^−d/2.

Main tool: Fourier Inversion Formula P(Sn= 0) = 1

(2π)^d Z

[−π,π]^d

E(e^iΘ·Sⁿ)dΘ Use that S_n is a sum of i.i.d. random vectors and for||Θ|| small,

E(e^iΘ·X¹) = 1−||Θ||²

2d +o(||Θ||²)

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Theorem

A simple random walk in Z^d is recurrent for d= 1 or 2, but is transient for d ≥3.

Another way to say that :

”All roads lead to Rome except the cosmic paths ! ”

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Simple Random Walks inZ^d Asymptotic distribution fornlarge

Local limit theorem

For n andx integers s.t. n+x is even,

P(Sn=x) = Number of paths of length n from 0 to x Number of paths of length n

= Cn^(n+x)/2

2ⁿ

∼ r2

π.e⁻^x

2

√2n

n for n large and |x|=o(n^2/3) Therefore, for any x∈Rs.t. n+ [x√

n] is even,

P(Sn= [x√ n]) ∼

r2 π.e⁻^x

2

√2

n for n large

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Let a,b ∈Rwith a<b, P

S_2n∈[a√ 2n,b√

2n]

= X

k∈[a√ 2n,b√

2n]

P(S_2n=k)

∼ r2

n

X

m∈[a,b]∩^√²^Z

2n

√1 2πe⁻^m

2 2

→ 1

√2π Z b

a

e⁻^x

2

2 dx =P(X ∈[a,b]) where X is distributed as the Normal distribution N(0,1).

Notation: As n tends to infinity, Sn

√n

−→ NL (0,1).

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Maximum of the path at time n

Define

M_n := max

k=0..nS_k

= max

t∈[0,1]S_[nt]

For anyt >0, as n large, ^S^√^[nt]_n ∼ N(0,t) and Mn

√n = max

t∈[0,1]

S_[nt]

√n

Functional of the path from 0 to time n, a convergence in distribution on the space of the c`ad-l`ag paths (φ(t))_t∈[0,1] is needed.

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Functional limit theorem

The sequence _S

√[nt]

n

t≥0 converges in law to the real Brownian motion (B_t)t≥0, that is a stochastic process satisfying :

B₀ := 0

Stationarity of the increments : B_t−B_s ∼Bt−s for s <t Independence of the increments : Bt−Bs independent from Bs

Bt ∼ N(0,t)

The law of the maximum of the Brownian motion is well-known :

t∈[0,1]max B_t∼B₁∼ N(0,1)

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Arcsine distributions

With the same method, we can compute the asymptotic distributions of many functionals of the random walk :

N_n= max{k = 1. . .n ;S_k = 0} the last time the random walker visits 0 before time n

V_n= #{k = 1. . .n ;S_k >0}the number of positive values of the random walk before time n

We have for any x∈(0,1), asn is large, P(N_n≤xn)∼ 2

π arcsin(√ x) and

P(V_n≤xn)∼ 2

πarcsin(√ x).

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Simple Random Walks inZ^d Asymmetric random walk

The random walker moves to the right with probability p and to the left with probability q = 1−p.

Same questions as before: Recurrence, Transience, Asymptotic distribution,....

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Let (X_i)_i≥1 be i.i.d. random variables taking values +1 or−1 with probability p andq = 1−p respectively. The position of the walker at time n is given by :

S0:= 0 and for any n≥1,

S_n:=

n

X

i=1

X_i From this writing, we can compute

E(X₁) =p−q 6= 0 The strong law of large numbers gives : as n→+∞,

Sn

n = 1 n

n

X

i=1

Xi →E(X1) =p−q a.s.

The random walk (S_n)_n is transient, tends to +∞ (resp. −∞) when p >q (resp. p<q).

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Asymptotic distribution for n large

As n→+∞,

S_n−n (p−q)

√n

−→ NL (0, σ²) whereσ² = 4p(1−p).

Indeed, forn integer andx∈Zs.t. n+x is even, P(Sn=x) = Cn^(n+x)/2p^(n+x)/2q^(n−x)/2

∼ ....

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Random Walks in Random Environments Definition

Random Environment : Letω_x,x∈Z, be i.i.d. random variables with values in [0,1], uniformly bounded away from 0 and 1.

For a given realization of the environment, we consider the Markov chain (S_n)_n which jumps to the sitex+ 1, with probabilityω_x and to x−1 with probability 1−ω_x, given it is located at x.

They were introduced by A.A. Chernov in 1967 in order to model the replication of DNA.

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Random Walks in Random Environments Definition

Quenched law: Denote by P_ω the law of the walk (starting from 0) in the environmentω.

Annealed law: IfPdenotes the law of the environment, P =P×P_ω

defined as

P(.) = Z

Pω(.)dP(ω) Fundamental remark :

Under P, the random walk isnota Markov chain.

(Under Pω, the random walk is a Markov chain (inhomogeneous in space))

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Random Walks in Random Environments Recurrence-Transience

Solomon I

The ratio

ρ_x := 1−ω_x ωx

plays an important role in the study of RWRE.

Theorem (Solomon (1975))

IfE(lnρ0)<0 (resp. >0) then the random walk istransientand

n→+∞lim S_n= +∞ (resp− ∞) P −a.s.

IfE(lnρ₀) = 0, then the random walk is recurrentand lim sup

n→+∞

S_n= +∞ and lim inf

n→+∞S_n=−∞ P −a.s.

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Random Walks in Random Environments Recurrence-Transience

Solomon II

Theorem (Solomon (1975)) P-almost surely,

n→+∞lim Sn

n =v where

v =







1−E(ρ0)

1+E(ρ0) if E(ρ0)<1

E(1/ρ0)−1

E(1/ρ0)+1 if 1<1/E(ρ⁻¹₀ )

0 if 1/E(ρ⁻¹₀ )≤1≤E(ρ₀) Comparison with the random walk in Z:

1-|v|<|E(S₁)| −→ some slowdown already occurs.

2- The random walk can be transient with zero speed !

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Random Walks in Random Environments Valleys (or traps) - Slowing down

Potential:

V(x) =











x

X

k=1

log(ρ_k) if x≥1

0 if x= 0

−

0

X

k=x+1

log(ρk) if x ≤ −1

(V(x))_x is a realrandom walk with mean E[logρ₀] and variance E[(logρ₀)²].

Remark also that

ω_x = e^−V^(x)

e^−V^(x−1)+e^−V^(x) > 1 2 if and only if

V(x−1)>V(x).

When the potential decreases (resp. increases), the random walker tends to go to the right (resp. left).

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Random Walks in Random Environments Asymptotic distributions fornlarge

Recurrent case – E [log ρ

₀

] = 0

Theorem (Sinai (1982), Kesten (1986), Golosov (1986)) Denote

σ² =E(logρ0)²∈ ]0,+∞[

Then,

σ²

(logn)²S_n−→^L b∞

where b∞ is a symmetric random variable with Laplace transform E(e^−λ|b^∞^|) = cosh(√

2λ)−1 λcosh(√

2λ) , λ >0.

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Transient case – E [log ρ

₀

] < 0

Under both assumptions :

1- There exists κ >0 s.t. E(ρ^κ₀) = 1 andE(ρ^κ₀(logρ0)+)<∞.

2- The distribution of log(ρ₀) is non-lattice.

Theorem (Kesten-Kozlov-Spitzer (1975)) Whenκ <1, (v = 0)

n→+∞lim PS_n n^κ ≤x

= 1−L_κ,b(x^−1/κ) Whenκ∈(1,2),

n→+∞lim P

Sn−nv v^1+1/κn^1/κ ≤x

= 1−L_κ,b(−x).

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Transient case – E [log ρ

₀

] < 0

L_κ,b is a stable distribution with characteristic function ˆLκ,b(t) = exp

−b|t|^κ

1−i t

|t|tan(πκ/2)

The value of b for κ∈ (0,2) was determined by Enriquez, Sabot and Zindy (’09)

Theorem (Kesten-Kozlov-Spitzer (1975)) Whenκ >2,

Sn−nv

√n

−→ NL (0, σ²) where σ²>0.

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Random Walk in Random Scenery

Riddle

Is this random walk recurrent or transient ?

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Theorem (Campanino – P´etritis (2003))

The random walk (S_n)_n is transient for almost every realization of the orientations.

A local limit theorem can even be proved.

Theorem (Castell – Guillotin-Plantard – P`ene – Schapira (AOP, 2011)) For n large,

P[S_n= 0]∼ C n^5/4.

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(Sn)n has the same distribution as (Xn,Yn)n where Y_n is the ”blue” random walk on Z.

Xn is the random walk (Yn) in random scenery (”H”,”T”) : ξi = 1 (resp. −1) if ”Tail” (resp. ”Head”) at site i ∈Z,

X_n =

n−1

X

k=0

ξ_Y_k

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We have

P[S_n= 0] = P[X_n= 0;Y_n= 0]

∼ P[Xn= 0|Y_n= 0]P[Yn= 0]

We know that

P[Yn= 0]∼ C n^1/2 and (not easy !)

P[X_n= 0|Y_n= 0]∼ C n^3/4