Introduction to random walks in random and non-random ...joye/quawagtalks/Guillotin.pdf · Simple Random Walks in Zd De nition Let (X i) i 1 be i.i.d. random variables taking values

Introduction to random walks in random andnon-random environments

Nadine Guillotin-Plantard

Institut Camille Jordan - University Lyon I

Grenoble – November 2012

Nadine Guillotin-Plantard (ICJ) Introduction to random walks in random and non-random environmentsGrenoble – November 2012 1 / 36

Outline

1 Simple Random Walks in Zd

DefinitionRecurrence - TransienceAsymptotic distribution for n largeAsymmetric random walk

2 Random Walks in Random EnvironmentsDefinitionRecurrence-TransienceValleys (or traps) - Slowing downAsymptotic distributions for n large

3 Random Walk in Random Scenery


Simple Random Walks in Zd 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”)





Natural questions

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

Mean position of the walker, fluctuations around this position, largedeviations...

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

Number of distinct sites visited by the walker up to time n (Range)

Maximal (or minimal) position of the walker before time n

Number of visits to a fixed site x . (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 walkerand so on...



Let (Xi )i≥1 be i.i.d. random variables taking values +1 or −1 with equalprobability.{Xi = +1} ={ The walker gets ”Head” at time i}.

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

S0 := 0

and for any n ≥ 1,

Sn :=n∑

i=1

Xi

(Sn)n≥0 is called simple random walk on Z.From this writing, we can compute

E (Sn) = 0

and Var(Sn) = E (S2n ) = n. Therefore, Sn ∼

√n


Simple Random Walks in Zd Recurrence - Transience

For n integer,

P(S2n = 0) =Number of paths of length 2n from 0 to 0

Number of paths of length 2n

=Cn

2n

22n

=(2n)!

4n(n!)2

∼ 1√πn

for n large

using Stirling’s formula

n! ∼(n

e

)n√2πn.



A random walk is said recurrent iff

P[

lim supn{Sn = 0}

]= P

[Sn = 0 i.o.

]= 1

Otherwise, it is called transient. Since Sn is a Markov chain, we have thisuseful criterion :

Theorem

(Sn)n is recurrent iff+∞∑n=0

P(Sn = 0) = +∞

Since P(Sn = 0) ∼ C/√

n, the simple random walk on Z is recurrent.



Simple random walk in Z2

Simple random walk in Z2



Georges Polya (1887 – 1985)



In higher dimension

Theorem (Polya (1921) )

There exists some constant C = C (d) s.t. for n large enough

P(Sn = 0) ∼ C n−d/2.

Main tool: Fourier Inversion Formula

P(Sn = 0) =1

(2π)d

∫[−π,π]d

E (e iΘ·Sn)dΘ

Use that Sn is a sum of i.i.d. random vectors and for ||Θ|| small,

E (e iΘ·X1) = 1− ||Θ||2

2d+ o(||Θ||2)



Theorem

A simple random walk in Zd is recurrent for d = 1 or 2, but is transientfor d ≥ 3.

Another way to say that :”All roads lead to Rome except the cosmic paths ! ”


Simple Random Walks in Zd Asymptotic distribution for n large

Local limit theorem

For n and x 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

=C

(n+x)/2n

2n

∼√

2

π.e−

x2

2n

√n

for n large and |x | = o(n2/3)

Therefore, for any x ∈ R s.t. n + [x√

n] is even,

P(Sn = [x√

n]) ∼√

2

π.e−

x2

2

√n

for n large



Let a, b ∈ R with a < b,

P(

S2n ∈ [a√

2n, b√

2n])

=∑

k∈[a√

2n,b√

2n]

P(S2n = k)

∼√

2

n

∑m∈[a,b]∩ 2Z√

2n

1√2π

e−m2

2

→ 1√2π

∫ b

ae−

x2

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

L−→ N (0, 1).



Maximum of the path at time n

Define

Mn := maxk=0..n

Sk

= maxt∈[0,1]

S[nt]

For any t > 0, as n large,S[nt]√

n∼ N (0, t) and

Mn√n

= maxt∈[0,1]

(S[nt]√

n

)Functional of the path from 0 to time n, a convergence in distribution onthe space of the cad-lag paths (φ(t))t∈[0,1] is needed.





Functional limit theorem

The sequence(S[nt]√

n

)t≥0

converges in law to the real Brownian motion

(Bt)t≥0, that is a stochastic process satisfying :

B0 := 0

Stationarity of the increments : Bt − Bs ∼ 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 :

maxt∈[0,1]

Bt ∼ B1 ∼ N (0, 1)



Arcsine distributions

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

Nn = max{k = 1 . . . n ; Sk = 0} the last time the random walkervisits 0 before time n

Vn = #{k = 1 . . . n ; Sk > 0} the number of positive values of therandom walk before time n

We have for any x ∈ (0, 1), as n is large,

P(Nn ≤ xn) ∼ 2

πarcsin(

√x)

and

P(Vn ≤ xn) ∼ 2

πarcsin(

√x).




Simple Random Walks in Zd Asymmetric random walk

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

Same questions as before: Recurrence, Transience, Asymptoticdistribution,....



Let (Xi )i≥1 be i.i.d. random variables taking values +1 or −1 withprobability p and q = 1− p respectively. The position of the walker attime n is given by :

S0 := 0

and for any n ≥ 1,

Sn :=n∑

i=1

Xi

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∑i=1

Xi → E (X1) = p − q a.s.

The random walk (Sn)n is transient, tends to +∞ (resp. −∞) whenp > q (resp. p < q).



Asymptotic distribution for n large

As n→ +∞,Sn − n (p − q)√

n

L−→ N (0, σ2)

where σ2 = 4p(1− p).

Indeed, for n integer and x ∈ Z s.t. n + x is even,

P(Sn = x) = C(n+x)/2n p(n+x)/2q(n−x)/2

∼ ....


Random Walks in Random Environments Definition

Random Environment : Let ωx , x ∈ Z, be i.i.d. random variables withvalues in [0, 1], uniformly bounded away from 0 and 1.For a given realization of the environment, we consider the Markov chain(Sn)n which jumps to the site x + 1, with probability ωx and to x − 1 withprobability 1− ωx , given it is located at x .

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


Random Walks in Random Environments Definition

Quenched law: Denote by Pω the law of the walk (starting from 0)in the environment ω.

Annealed law: If P denotes the law of the environment,

P = P× Pω

defined as

P(.) =

∫Pω(.) dP(ω)

Fundamental remark :Under P, the random walk is not a Markov chain.(Under Pω, the random walk is a Markov chain (inhomogeneous in space))


Random Walks in Random Environments Recurrence-Transience

Solomon I

The ratio

ρx :=1− ωx

ωx

plays an important role in the study of RWRE.

Theorem (Solomon (1975))

If E(ln ρ0) < 0 (resp. > 0) then the random walk is transient and

limn→+∞

Sn = +∞ (resp−∞) P − a.s.

If E(ln ρ0) = 0, then the random walk is recurrent and

lim supn→+∞

Sn = +∞ and lim infn→+∞

Sn = −∞ P − a.s.


Random Walks in Random Environments Recurrence-Transience

Solomon II

Theorem (Solomon (1975))

P-almost surely,

limn→+∞

Sn

n= v

where

v =

1−E(ρ0)1+E(ρ0) if E(ρ0) < 1E(1/ρ0)−1E(1/ρ0)+1 if 1 < 1/E(ρ−1

0 )

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 !


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

Potential:

V (x) =

x∑k=1

log(ρk) if x ≥ 1

0 if x = 0

−0∑

k=x+1

log(ρk) if x ≤ −1

(V (x))x is a real random walk with mean E[log ρ0] and varianceE[(log ρ0)2].Remark also that

ωx =e−V (x)

e−V (x−1) + e−V (x)>

1

2

if and only ifV (x − 1) > V (x).

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






Random Walks in Random Environments Asymptotic distributions for n large

Recurrent case – E[log ρ0] = 0

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

Denoteσ2 = E(log ρ0)2 ∈ ]0,+∞[

Then,σ2

(log n)2Sn

L−→ b∞

where b∞ is a symmetric random variable with Laplace transform

E(e−λ|b∞|) =cosh(

√2λ)− 1

λ cosh(√

2λ), λ > 0.



Transient case – E[log ρ0] < 0

Under both assumptions :1- There exists κ > 0 s.t. E(ρκ0) = 1 and E(ρκ0(log ρ0)+) <∞.2- The distribution of log(ρ0) is non-lattice.

Theorem (Kesten-Kozlov-Spitzer (1975))

When κ < 1, (v = 0)

limn→+∞

P(Sn

nκ≤ x

)= 1− Lκ,b(x−1/κ)

When κ ∈ (1, 2),

limn→+∞

P( Sn − nv

v 1+1/κn1/κ≤ x

)= 1− Lκ,b(−x).



Transient case – E[log ρ0] < 0

Lκ,b is a stable distribution with characteristic function

Lκ,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 andZindy (’09)

Theorem (Kesten-Kozlov-Spitzer (1975))

When κ > 2,Sn − nv√

n

L−→ N (0, σ2)

where σ2 > 0.


Random Walk in Random Scenery

Riddle

Is this random walk recurrent or transient ?



Theorem (Campanino – Petritis (2003))

The random walk (Sn)n is transient for almost every realization of theorientations.

A local limit theorem can even be proved.

Theorem (Castell – Guillotin-Plantard – Pene – Schapira (AOP, 2011))

For n large,

P[Sn = 0] ∼ C

n5/4.



(Sn)n has the same distribution as (Xn,Yn)n where

Yn 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,

Xn =n−1∑k=0

ξYk



We have

P[Sn = 0] = P[Xn = 0; Yn = 0]

∼ P[Xn = 0|Yn = 0]P[Yn = 0]

We know that

P[Yn = 0] ∼ C

n1/2

and (not easy !)

P[Xn = 0|Yn = 0] ∼ C

n3/4


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Introduction to random walks in random and non-random ...joye/quawagtalks/Guillotin.pdf · Simple Random Walks in Zd De nition Let (X i) i 1 be i.i.d. random variables taking values