Annealed and quenched ﬂuctuations of transient random walks in random environment on Z

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Annealed and quenched fluctuations of transient random walks in random environment

on Z

Laurent Tournier

UMPA, ENS Lyon – IMPA, Rio de Janeiro Joint work with N. Enriquez, C. Sabot and O. Zindy

June 16, 2011. LaPietra Week in Probability, Florence

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RWRE on Z : definitions

Letµbe a law on(0,1).

Define an i.i.d. sequenceω= (ωx)x∈Zwith lawµ(“environment”).

ω0 ω1

ω₋1

1−ω₋1

0 1 2 3

− 1

− 2

− 3 · · · · · ·

ex.:µ=pδα+ (1−p)δβ, µ=B(a,b)(Beta distribution) (Givenω) QuenchedlawPωof Markov chain of transitionω (Randomω) AnnealedlawPof RWRE:

P(·) =E[Pω(·)] = Z

Pω(·)dµ^⊗^Z(ω) UnderP, the RW “learns” aboutω; transitions arereinforced.

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RWRE on Z : definitions

ω0 ω1

ω−1

1−ω1

1−ω0

1−ω−1

0 1 2 3

− 1

− 2

− 3 · · · · · ·

· · ·

ex.:µ=pδα+ (1−p)δβ, µ=B(a,b)(Beta distribution) (Givenω) QuenchedlawP_ωof Markov chain of transitionω (Randomω) AnnealedlawPof RWRE:

P(·) =E[Pω(·)] = Z

P_ω(·)dµ^⊗^Z(ω) UnderP, the RW “learns” aboutω; transitions arereinforced.

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RWRE on Z : definitions

ω0 ω1

ω−1

1−ω1

1−ω0

1−ω−1

0 1 2 3

− 1

− 2

− 3 · · · · · ·

· · ·

ex.:µ=pδα+ (1−p)δβ, µ=B(a,b)(Beta distribution)

(Givenω) QuenchedlawP_ωof Markov chain of transitionω (Randomω) AnnealedlawPof RWRE:

P(·) =E[Pω(·)] = Z

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RWRE on Z : definitions

ω0 ω1

ω−1

1−ω1

1−ω0

1−ω−1

0 1 2 3

− 1

− 2

− 3 · · · · · ·

· · ·

P(·) =E[Pω(·)] = Z

P_ω(·)dµ^⊗^Z(ω)

UnderP, the RW “learns” aboutω; transitions arereinforced.

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RWRE on Z : definitions

ω0 ω1

ω−1

1−ω1

1−ω0

1−ω−1

0 1 2 3

− 1

− 2

− 3 · · · · · ·

· · ·

P(·) =E[Pω(·)] = Z

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Potential – Transience and speed

0 1 2 3

− 1

− 2

− 3 ω

DefineVbyV(0) =0 andωx= e^−V^(x) e^−V(x)+e^−V(x−1),

e^−V(x)=C_x,x+1

i.e.V(x) =logρ1+· · ·+logρxforx≥0, whereρx= 1−ωx

ωx

Theorem (Solomon 1975)

( X_n→+∞ iff E[logρ0]<0(⇔V(x)_x→∞−→ −∞) (Xn)nrecurrent iff E[logρ0] =0

AssumeE[logρ0]<0.P-a.s.,Xn

n →vwhere

v>0 if E[ρ0]<1 v=0 else.

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Potential – Transience and speed

0 1 2 3

− 1

− 2

− 3 V ω

V (x) < V (x − 1) ⇔ ω

_x

>

¹₂

e^−V(x)=C_x,x+1

i.e.V(x) =logρ1+· · ·+logρxforx≥0, whereρx= 1−ωx

ωx

n →vwhere

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Potential – Transience and speed

0 1 2 3

− 1

− 2

− 3 V ω

V (x) < V (x − 1) ⇔ ω

_x

>

¹₂

e^−V(x)=C_x,x+1

i.e.V(x) =logρ1+· · ·+logρxforx≥0, whereρx=1−ωx

ωx

n →vwhere

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Potential – Transience and speed

0 1 2 3

− 1

− 2

− 3 V ω

V (x) < V (x − 1) ⇔ ω

_x

>

¹₂

DefineVbyV(0) =0 andωx= e^−V^(x)

e^−V(x)+e^−V(x−1),e^−V(x)=Cx,x+1

ωx

n →vwhere

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Potential – Transience and speed

0 1 2 3

− 1

− 2

− 3 V ω

V (x) < V (x − 1) ⇔ ω

_x

>

¹₂

e^−V(x)+e^−V(x−1),e^−V(x)=Cx,x+1

ωx

P-a.s.,

( Xn→+∞ iff E[logρ0]<0(⇔V(x)_x→∞−→ −∞) (Xn)nrecurrent iff E[logρ0] =0

n →vwhere

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Potential – Transience and speed

0 1 2 3

− 1

− 2

− 3 V ω

V (x) < V (x − 1) ⇔ ω

_x

>

¹₂

e^−V(x)+e^−V(x−1),e^−V(x)=Cx,x+1

ωx

Pω-a.s.

for P-a.e.ω,

( Xn →+∞ iff E[logρ0]<0(⇔V(x)_x→∞−→ −∞) (Xn)nrecurrent iff E[logρ0] =0

n →vwhere

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Potential – Transience and speed

0 1 2 3

− 1

− 2

− 3 V ω

V (x) < V (x − 1) ⇔ ω

_x

>

¹₂

e^−V(x)+e^−V(x−1),e^−V(x)=Cx,x+1

ωx

Pω-a.s.

for P-a.e.ω,

( Xn →+∞ iff E[logρ0]<0(⇔V(x)_x→∞−→ −∞) (Xn)nrecurrent iff E[logρ0] =0

n →vwhere

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Fluctuations: KKS theorem

Hypotheses

(a)∃κ >0 such thatE[ρ^κ₀] =1, andE[ρ^κ₀(logρ0)+]<∞ (b) The law of logρ0is non-arithmetic.

Ex. Forµ=B(a,b),κ=b−a.

Theorem (Kesten-Kozlov-Spitzer 1975) Assume (a)-(b). Then, underP,

If 0< κ <1, Xn

n^κ

(law)

−→_n (AκS^κ)^−1/κ E[e^itS^κ] =e^−(−it)^κ If 1< κ <2, X_n−vn

n^1/κ

(law)

−→_n −v¹⁺^κ¹A_κS^κ E[e^itS^κ] =e^(−it)^κ Ifκ >2, X_n−vn

√n

(law)

−→_n N(0, σ²)

whereAκ>0 (S^κis a totally asymmetricκ-stable r.v.)

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Fluctuations: KKS theorem

Hypotheses

(a)∃κ >0 such thatE[e^κ^log^ρ⁰] =1, andE[ρ^κ₀(logρ0)+]<∞ (b) The law of logρ0is non-arithmetic.

Theorem (Kesten-Kozlov-Spitzer 1975) Assume (a)-(b). Then, underP,

If 0< κ <1, Xn

n^κ

(law)

−→_n (AκS^κ)^−1/κ E[e^itS^κ] =e^−(−it)^κ If 1< κ <2, Xn−vn

n^1/κ

(law)

−→_n −v¹⁺^κ¹A_κS^κ E[e^itS^κ] =e^(−it)^κ Ifκ >2, Xn−vn

√n

(law)

−→_n N(0, σ²)

whereAκ>0 (S^κis a totally asymmetricκ-stable r.v.)

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Fluctuations: KKS theorem

Hypotheses

(a)∃κ >0 such thatE[e^κ^log^ρ⁰] =1, andE[ρ^κ₀(logρ0)+]<∞ (b) The law of logρ0is non-arithmetic.

Theorem (Kesten-Kozlov-Spitzer 1975)

Assume (a)-(b). Then, underP, (letτ(x) =inf{n:X_n=x}) If 0< κ <1, Xn

n^κ

(law)

−→_n (AκS^κ)^−1/κ τ(x) x^1/κ

(law)

−→_n AκS^κ If 1< κ <2, Xn−vn

n^1/κ

(law)

−→_n −v¹⁺^κ¹AκS^κ τ(x)−^xv

x^1/κ

(law)

−→_x AκS^κ Ifκ >2, X_n−vn

√n

(law)

−→_n N(0, σ²) τ(x)√−^xv

x

(law)

−→_x N(0, σ⁰²) whereAκ>0 (S^κis a totally asymmetricκ-stable r.v.)

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Main result

Theorem (Enriquez-Sabot-T.-Zindy 2010) For 0< κ <2 (κ6=1),

Aκ=2

πκ²

|sin(πκ)|(CK)²E[ρ^κ₀logρ0] ^1/κ

whereCKis Kesten’s renewal constant:P(R>t)∼CKt^−κwith R=1+ρ1+ρ1ρ2+· · ·.

Description of thequenchedbehaviour for 0< κ <2

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Main result

Theorem (Enriquez-Sabot-T.-Zindy 2010) For 0< κ <2 (κ6=1),

Aκ=2

πκ²

|sin(πκ)|(CK)²E[ρ^κ₀logρ0] ^1/κ

whereCKis Kesten’s renewal constant:P(R>t)∼CKt^−κwith R=1+ρ1+ρ1ρ2+· · ·.

Description of thequenchedbehaviour for 0< κ <2

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Sample trajectory (0 < κ < 1)

150

40000 100

20000 10000

0 50000 60000

0 50

30000 70000

200 250

-50

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Sample trajectory (1 < κ < 2)

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Sample trajectory (1 < κ < 2)

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General scheme of proof

Excursions ofV:e0=0,en+1 =inf{k≥en|V(k)≤V(en)}. V

τ(en) =X

k

τ(ek+1)−τ(ek)

=

small exc. H<hn

+

high exc. H≥hn

•There are very few large excursions (⇒crossing times almost i.i.d.)

•

Crossings of small excursions iso(n^1/κ) (0< κ <1)

Fluctuation of crossings of small excursions iso(n^1/κ) (1< κ <2)

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General scheme of proof

Hk= max

e_k≤x<e_k+1V(x)−V(ek) ek

τ(en) =X

k

τ(ek+1)−τ(ek)

=

small exc. H<hn

+

high exc. H≥hn

•

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General scheme of proof

Hk= max

e_k≤x<e_k+1V(x)−V(ek) ek

τ(en) =X

k

τ(ek+1)−τ(ek)

=

small exc.

H<hn

+

high exc.

H≥hn

•

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Crossing time of a (high) excursion

Goal: estimateP(τ(e1)>t)ast→ ∞.

P(τ(e1)>t) =P(τ(e1)>t,H>logt−log logt) +o(t^−κ)

F

_i

V

H 0

S

e

₁

•M1,M2,Hare almost independent on{H>h}withhlarge

•Property (Feller – Iglehart):P(e^H>u)∼CIu^−κ Thus,P(τ(e1)>t)∼Ct^−κfor some (rather explicit)C.

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Crossing time of a (high) excursion

F

_i

V

H 0

S

e

₁

τ(e1) =F1+· · ·+FN

'M1M2e^He e∼ E(1),

M1=PTH

−∞e^−V(x),M2=P∞ 0 e^V(x)−H

•Property (Feller – Iglehart):P(e^H>u)∼CIu^−κ Thus,P(τ(e1)>t)∼Ct^−κfor some (rather explicit)C.

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Crossing time of a (high) excursion

F

_i

V

H 0

S

e

₁

τ(e1) =F1+· · ·+FN+S

'M1M2e^He e∼ E(1),

M1=PTH

−∞e^−V(x),M2=P∞ 0 e^V(x)−H

•M₁,M₂,Hare almost independent on{H>h}withhlarge

•Property (Feller – Iglehart):P(e^H>u)∼C_Iu^−κ Thus,P(τ(e1)>t)∼Ct^−κfor some (rather explicit)C.

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Crossing time of a (high) excursion

F

_i

V

H 0

S

e

₁

τ(e1) =F1+· · ·+FN+S 'Eω[F]N

'M1M2e^He e∼ E(1),

M1=PTH

−∞e^−V(x),M2=P∞ 0 e^V(x)−H

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Crossing time of a (high) excursion

F

_i

V

H 0

S

e

₁

τ(e1) =F1+· · ·+FN+S 'Eω[F]N

'Eω[F]Eω[N]e

'M1M2e^He

e∼ E(1),

M1=PTH

−∞e^−V(x),M2=P∞ 0 e^V(x)−H

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Crossing time of a (high) excursion

F

_i

V

H 0

S

e

₁

M

1

M

₂

T

_H

τ(e1) =F1+· · ·+FN+S 'Eω[F]N

'Eω[F]Eω[N]e 'M1M2e^He e∼ E(1),

M1=PTH

−∞e^−V(x),M2=P∞ 0 e^V(x)−H

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Interlude on heavy-tailed distributions

LetT1,T2, . . .be i.i.d. r.v.≥0 such that P(Ti>t)∼Ct^−κ. Then, if 0< κ <1, T₁+· · ·+T_n

n^1/κ

(law)

−→_n (CΓ(1−κ))^1/κS^κ and, if 1< κ <2, T1+· · ·+Tn−nE[Ti]

n^1/κ

(law)

−→_n (−CΓ(1−κ))^1/κS^κ. Forκ >2, CLT.

Heavy-tail phenomenon:

•#{1≤i≤n:T_i≥εn^1/κ}^(law)−→_n P(Cε^−κ)

•(for 0< κ <1)E h X

1≤i≤n

Ti1_{T

i<εn^1/κ}

i∼Cε^1−κn^1/κ

⇒Up to an error of orderε^1−κ,^T¹^+···+T_n_1/κ ⁿ is given by theP(Cε^−κ)terms larger thanεn^1/κ.

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Interlude on heavy-tailed distributions

LetT1,T2, . . .be i.i.d. r.v.≥0 such that P(Ti>t)∼Ct^−κ. Then, if 0< κ <1, T1+· · ·+T_n

n^1/κ

(law)

−→_n (CΓ(1−κ))^1/κS^κ and, if 1< κ <2, T1+· · ·+Tn−nE[Ti]

n^1/κ

(law)

−→_n (−CΓ(1−κ))^1/κS^κ. Forκ >2, CLT.

Heavy-tail phenomenon:

•#{1≤i≤n:T_i≥εn^1/κ}^(law)−→_n P(Cε^−κ)

•(for 1< κ <2) Var X

1≤i≤n

Ti1_{T_i_<εn1/κ}

∼Cε^2−κn^2/κ

⇒Up to an error of orderε^1−κ/2, ^T¹^+···+T_n1/κⁿ^−nE[Tⁱ^] is given by the (centered) P(Cε^−κ)terms larger thanεn^1/κ.

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Next steps of the proof of the main result (KKS)

Lethn=_κ¹ logn−log logn(hencee^hⁿ=εnn^1/κ,εn= _log¹_n, cf.τ'Me^H)

τ(en) =X

k

τ(ek+1)−τ(ek)

=

small exc.

H<hn

+

high exc.

H≥hn

Neglect (fluctuations of) crossing times of small excursions Ensure large excursions are way appart of each other w.h.p.

Neglect time spent “backtracking” far away to the left of a high excursion before crossing it

Replace neglected parts by independent versions of them (which are negligible as well)

⇒Reduction to i.i.d. copies ofτ(e1), hence the (annealed) limit theorem.

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Next steps of the proof of the main result (KKS)

Lethn=_κ¹ logn−log logn(hencee^hⁿ=εnn^1/κ,εn= _log¹_n, cf.τ'Me^H)

τ(en) =X

k

τ(ek+1)−τ(ek)

=

small exc.

H<hn

+

high exc.

H≥hn

Neglect (fluctuations of) crossing times of small excursions Ensure large excursions are way appart of each other w.h.p.

Neglect time spent “backtracking” far away to the left of a high excursion before crossing it

Replace neglected parts by independent versions of them (which are negligible as well)

⇒Reduction to i.i.d. copies ofτ(e1), hence the (annealed) limit theorem.

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Consequences of the quenched description

0< κ <1

τ(en)is mainly given by a few termsMe^Heattached to deep valleys.

The time spent in-between is negligible in comparison to them.

⇒Localizationin a (random) deep valley. (cf. Enriquez-Sabot-Zindy) 1< κ <2

The fluctuations ofτ(en)are mainly given by a few termsMe^H(e−1) attached to deep valleys.

⇒Practical interest(explicit confidence intervals,. . . )

The fluctuations ofτ(en)are almost a sum of i.i.d. terms likeMe^H(e−1) (re-introducing independent small valleys).

L

τ(en)−E_ω[τ(en)]

n^1/κ ω

' L

n

X

i=1

Mb_ie^b^Hⁱ

n^1/κ (ei−1)

M,b Hb

!

⇒Limit theoremfor the law of the quenched law of fluctuations IfT1,T2, . . .are i.i.d. withP(T >t)∼Ct^−κand 0< κ <2,

T_i n^1/κ

1≤i≤n (law)

−→ {ξi|i≥1} whereξis a PPP of intensityλκt^−(κ+1)dt.

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Consequences of the quenched description

0< κ <1

τ(en)is mainly given by a few termsMe^Heattached to deep valleys.

The time spent in-between is negligible in comparison to them.

⇒Localizationin a (random) deep valley. (cf. Enriquez-Sabot-Zindy) 1< κ <2

The fluctuations ofτ(en)are mainly given by a few termsMe^H(e−1) attached to deep valleys.

⇒Practical interest(explicit confidence intervals,. . . )

The fluctuations ofτ(en)are almost a sum of i.i.d. terms likeMe^H(e−1) (re-introducing independent small valleys).

L

τ(en)−Eω[τ(en)]

n^1/κ ω

' L

n

X

i=1

Mbie^b^Hⁱ

n^1/κ (ei−1)

M,b Hb

!

⇒Limit theoremfor the law of the quenched law of fluctuations L

τ(en)−Eω[τ(en)]

n^1/κ ω

(law,W1)

−→_n L

∞

X

i=1

ξi(ei−1) ξ

!

whereξis a PPP of intensityλκt^−(κ+1)dt. AndW1is Wasserstein distance.