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El e c t ro n ic

Jo ur

n a l o

f P

r o

b a b i_{l i t y} Vol. 16 (2011), Paper no. 6, pages 174–215.

Journal URL

http://www.math.washington.edu/~ejpecp/

A central limit theorem for random walk in a random

environment on marked Galton-Watson trees

Gabriel Faraud

∗

Abstract

Models of random walks in a random environment were introduced at first by Chernoff in 1967 in order to study biological mechanisms. The original model has been intensively studied since then and is now well understood. In parallel, similar models of random processes in a random environment have been studied. In this article we focus on a model of random walk on random marked trees, following a model introduced by R. Lyons and R. Pemantle (1992). Our point of view is a bit different yet, as we consider a very general way of constructing random trees with random transition probabilities on them. We prove an analogue of R. Lyons and R. Pemantle’s recurrence criterion in this setting, and we study precisely the asymptotic behavior, under re-strictive assumptions. Our last result is a generalization of a result of Y. Peres and O. Zeitouni (2006) concerning biased random walks on Galton-Watson trees.

Key words: Random Walk, random environment, tree, branching random walk, central limit theorem.

AMS 2000 Subject Classification:Primary 60K37; 60F05; 60J80.

Submitted to EJP on July 13, 2010, final version accepted December 22, 2010.

∗_{Laboratoire Analyse, Géométrie et Applications, UMR 7539, Institut Galilée, Université Paris 13, 99 avenue J.B.}

Clément 93430 Villetaneuse FRANCE. Research partially supported by the ANR project MEMEMO. faraud@math.univ-paris13.fr

1

Introduction and statement of results.

Models of random walks in a random environment were introduced at first by Chernov in 1967 ([6]) in order to study biological mechanisms. The original model has been intensively studied since then and is now well understood. On the other hand, more recently, several attempts have been made to study extensions of this original model, for example in higher dimensions, continuous time, or different space.

It is remarkable that the random walk inZd,d>1, is still quite mysterious, in particular no precise criterion for recurrence/transience has ever been found.

In the case of trees, however, a recurrence criterion exists, and even estimates for the asymptotic behavior have been proven. To present our model and the existing results, we begin with some notations concerning trees. LetT be a tree rooted at some vertexe. For each vertex x ofT we call

N(x)the number of his children_{x₁,x₂, ...,x_N(x)}, and←x−his father. For two vertices x,y _∈T, we calld(x,y)the distance between x and y, that is the number of edges on the shortest path from x

to y, and |x|:= d(e,x). LetT_n be the set of vertices such that|x|= n, and T∗ =T\ {e}. We also note x< y whenx is an ancestor of y.

We call a marked tree a couple(T,A), whereAis a random application from the vertices ofT toR∗

+.

LetT_{be the set of marked trees. We introduce the filtrationGnon}T_{defined as}

Gn=σ{N(x),A(xi), 1≤i≤n,|x|<n,x ∈T}. Following[20], given a probability measureqonN_⊗R∗N∗

+ , there exists a probability measureMTon

T_{such that}

• the distribution of the random variable(N(e),A(e₁),A(e₂), ...)isq,

• givenGn, the random variables(N(x),A(x1),A(x2), ...), forx ∈Tn, are independent and their conditional distribution isq.

We will always assumem:=E[N(e)]>1, ensuring that the tree is infinite with a positive probability. We now introduce the model of random walk in a random environment. Given a marked treeT, we set for x_∈T∗,x_i a child ofx,

ω(x,xi) =

A(x_i) 1+PN_j=(₁x)A(x_j) and

ω(x←−x ) = 1 1+PN_j=(₁x)A(x_j)

. Morever we setω(x,y) =0 wheneverd(x,y)₆=1,

It is easy to check that(ω(x,y))_x,y∈T is a family of non-negative random variables such that,

∀x _∈T,X y∈T

ω(x,y) =1, and

∀x _∈T∗,A(x) = ω(

←_x−_,x)

ω(←−x,←←x−−)

whereω(e,←−e)is artificially defined as 1

ω(e,←−e) =

X |x|=1

A(x).

Further,ω(x,y)₆=0 wheneverx and y are neighbors.

T will be called “the environment”, and we call “random walk on T” the Markov chain (X_n,P_T) defined byX0=eand

∀x,y _∈T, P_{T(Xn+1= y|Xn= x) =ω(x,y).} We call “annealed probability” the probabilityP

MT=MT⊗PT taking into account the total alea. We set, forx_∈T,C_x=Q_e<z≤xA(z). We can associate to the random walkX_nan electrical network with conductance Cx along[←x−,x], and a capacited network with capacity Cx along [←x−,x]. We recall the definition of an electrical current on an electrical network. LetG = (V,E)be a graph, C

be a symmetric function onE, andA,Zbe two disjoint subsets ofV. We define the electrical current betweenAandZas a functionithat is antisymmetric onEsuch that, for anyx ∈V\(A∪Z), the sum on the edgesestarting from x ofi(e)equals zero (this is call Kirchhoff’s node Law), and, moreover,

isatisfies the Kirchhoff’s cycle Law, that is, for any cycle x₁,x₂, . . . ,x_n=x₁, n

X

i=1

i(x_i,x_i+1)

C(x_i,x_i+1)=0.

A flow on a capacited network is an antisymmetric functionθthat satisfies the Kirchhoff’s node Law, and such that, for all edgese,θ(e)<C(e), (for more precisions on this correspondence we refer to the chapters 2 and 3 of[17]).

We shall also frequently use the convex functionρdefined forα≥0 as

ρ(α) =EMT

 N

(e)

X 0

A(ei)α

 =Eq

 XN

0

A(i)α

 .

Remark : This model is in fact inspired by a model introduced in[16]. In this case the tree T and theA(x)were introduced separately, and theA(x)were supposed to be independent. Here we can include models in which the structure of the tree and the transition probabilities are dependent. A simple example that is covered in our model is the following : Let T be a Galton-Watson tree. We chose an i.i.d. family (B(x))_x∈T and set, for every x _∈T, 1_≤ i_≤ N(x), A(x_i) = B(x). This way the transition probabilities to the children of any vertex are all equal, but randomly chosen. In R. Lyons and R. Pemantle’s article, a recurrence criterion was shown, our first result is a version of this criterion in our setting.

Theorem 1.1. We suppose that there exists0≤ α≤1such that ρis finite in a small neighborhood of α, ρ(α) =inf0≤t≤1ρ(t):= p and ρ′(α) = Eq

hPN(e)

i=1 A(ei)αlog(A(ei))

i

is finite. We assume that

PN(e)

i=1 A(ei)is not identically equal to1.

Then,

1. if p<1then the RWRE is a.s. positive recurrent, the electrical network has zero conductance a.s.,

2. if p_≤1then the RWRE is a.s. recurrent, the electrical network has zero conductance a.s. and the capacited network admits no flow a.s..

3. if p>1, then, given non-extinction, the RWRE is a.s. transient, the electrical network has positive

conductance a.s. and the capacited network admits flow a.s..

(By “almost surely” we mean “forMTalmost every T ”).

Remark: In the case wherePN_i=(₁e)A(e_i)is identically equal to 1, which belongs to the second case,

|Xn| is a standard unbiased random walk, therefore Xn is null recurrent. However, there exists a flow, given byθ(←−x,x) =Cx.

The proof of this result is quite similar to the proof of R. Lyons and R. Pemantle, but there are some differences, coming from the fact that in their settingi.i.d. random variables appear along any ray of the tree, whereas it is not the case here. Results on branching processes will help us address this problem.

Theorem 1.1 does not give a full answer in the casep=1, but this result can be improved, provided some technical assumptions are fulfilled. We introduce the condition

(H1): _∀α∈[0, 1], E_q

 

 N

(e)

X 0

A(e_i)α

 log+

 N

(e)

X 0

A(e_i)α

   <∞, In the critical case, we have the following

Proposition 1.1. We suppose p = 1, m > 1 and (H1). We also suppose that ρ′(1) =

E_qhPN_i=(₁e)A(e_i)log(A(e_i))iis defined and thatρis finite in a small neighborhood of1. Then,

• ifρ′(1)<0, then the walk is a.s. null recurrent, conditionally on the system’s survival,

• ifρ′(1) =0and for someδ >0,

E_MT[N(e)1+δ]<∞,

then the walk is a.s. null recurrent, conditionally on the system’s survival,

• ifρ′(1)>0,and if for someη >0,ω(x,←−x )> ηalmost surely, then the walk is almost surely

positive recurrent.

Remark: The distinction between the caseρ′(1) =0 andρ′(1)>0 is quite unexpected.

The study of the critical case turns out to be quite interesting, indeed several different behaviors appear in this case. The quantityκ=inf_{t>1,ρ(t)>1_}, associated toqis of particular interest. Whenρ′(1)_≥0, for regular trees and identically distributed A(x), Y. Hu and Z. Shi showed ([9]) that there exist constants 0<c_1≤c₂<∞such that

c1≤lim inf_n →∞

max_0<s<n|X_s|

(logn)3 ≤lim sup

n→∞

max_0<s<n|X_s|

(logn)3 ≤c2, P−a.s..

It was recently proven by G. Faraud, Y. Hu and Z. Shi that max0<s<n|Xs|

(logn)3 actually converges to an

explicit constant (see [7]). Interestingly, this constant has a different form whenρ′(1) = 0 and whenρ′(1)>0.

1 1 1

ρ(t) ρ(t) ρ(t)

t t t

ρ′(1)<0 ρ′(1) =0 ρ′(1)>0

1 κ 1 1

Figure 1: Possible shapes forρin the critical case In the caseρ′(1)<0, Y. Hu and Z. Shi showed ([8]) that

lim n→∞

log max_0<s<n|Xs| logn =1−

1

min_{κ, 2_}, P−a.s..

Results in the casep<1 have also been obtained by Y. Hu and Z. Shi ([8]), and the casep>1 has been studied by E. Aidekon ([1]).

Let us go back to the critical case. Our aim is to study what happens whenκis large. Whenκ≥2, the walk behaves asymptotically like n12. Our aim is to get a more precise estimate in this case.

However we are not able to cover the whole regimeκ∈[2,_∞]. We first introduce the ellipticity assumptions

∃0< ǫ₀<∞;∀i,ǫ₀≤A(e_i)_≤ 1

ǫ0

,q−a.s. (2)

and we assume that(A(e_i))_1≤i≤N(e)is of the form(A′(i)_1(i≤N(e))_i≥1, where(A′(i))_i≥1is a i.i.d. family independent ofN(e)and that E_q[N(e)κ+1]<∞. (H2)

Remark :We actually only need this assumption to show Lemma 4.3, we can, for example, alterna-tively suppose that

∃N₀;N(e)_≤N₀,q₋p.s. etP_q[N_≥2_|A(e₁)]_≥ 1

N0

(3) Note furthermore that those conditions imply (H1).

Theorem 1.2. Suppose N(e)_≥1, q−a.s., (2), (3).

If p = 1, ρ′(1) <0 and κ∈(8,_∞], then there is a deterministic constant σ >0 such that, forMT

almost every tree T, the process{|X_⌊nt⌋|/

p

σ2_{n}converges in law to the absolute value of a standard}

Brownian motion, as n goes to infinity.

Remark :This result is a generalization of a central limit theorem proved by Y. Peres and O. Zeitouni [21]in the case of a biased standard random walk on a Galton-Watson tree. In this case,A(x)is a constant equal to _m1, thereforeκ=_∞. Our proof follows the same lines as theirs.

In the annealed setting, things happen to be easier, and we can weaken the assumption onκ. Theorem 1.3. Suppose N(e)_≥1, q−a.s., (2), (3). If p=1,ρ′(1)<0andκ∈(5,∞], then there is

a deterministic constantσ >0such that, underP_{MT, the process{|X⌊nt⌋|/}

p

σ2_{n}converges in law to}

Remark : As we will see, the annealed CLT will even be true for κ ∈(2,_∞), on a different kind of tree, following a distribution that can be described as “the invariant distribution” for the Markov chain of the “environment seen from the particle”.

We thank P. Mathieu for indicating to us the technique of C. Kipnis and S.R.S. Varadhan ([12]), that was quite an inspiration for us.

Our article will be organized as follows

• In section 2 we show Theorem 1.1.

• In section 3 we introduce a new law on trees, with particular properties.

• In section 4 we show a Central Limit Theorem for random walks on trees following the “new law”.

• In section 5 we expose a coupling between the original law and the new one.

• In section 6 we show some lemmas.

• In section 7 we show Theorem 1.3

2

Proof of Theorem 1.1.

Let us first introduce an associated martingale, which will be of frequent use in the sequence. Letα∈R+_and

Y_n(α)= X

x∈Tn Y

e<z≤x

A(z)α= X

x∈Tn

Cα_x.

Y_n(α)is known as Mandelbrot’s Cascade. It it is easy to see that ifρ(α)<∞then Y

(α)

n

ρ(α)n is a non-negative martingale, with a.s. limitY

(α)_.

We have the following theorem, due to J.D. Biggins (1977) (see[3, 4]) that allows us to know when

Y(α)is non trivial.

Statement 2.1(Biggins). Letα∈R+_{. Supposeρ is finite in a small neighborhood ofα, and ρ}′(α)

exists and is finite, then the following are equivalent

• given non-extinction, Y(α)>0a.s.,

• P_MT[Y(α)=0]<1,

• E_MT[Y(α)] =1,

• E_qhP₀N(e)A(e_i)αlog+PN₀(e)A(e_i)αi<∞, and (H2):=αρ′(α)/ρ(α)<logρ(α),

This martingale is related to some branching random walk, and has been intensively studied ([18, 3, 4, 13, 14, 19]). We will see that it is closely related to our problem.

Let us now prove Theorem 1.1. We shall use the following lemma, whose proof is similar to the proof presented in page 129 of[16]and omitted.

Lemma 2.1.

min

0≤t≤1E 

X

x∈T1 A(x)t



= max

0<y≤1tinf>0y 1−t_E



X

x∈T1 A(x)t

 .

(1) Let us begin with the subcritical case, We suppose there exists some 0 < α < 1 such that

ρ(α) = inf_0≤t<1ρ(t) < 1. Then, following [11] (Prop 9-131), and standard electrical/capacited

network theory, if the conductances have finite sum, then the random walk is positive recurrent, the electrical network has zero conductance a.s., and the capacited network admits no flow a.s.. We have

X

x∈T∗

C_xα= ∞ X

n=0 X

x∈Tn

Cα_x =X

n

ρ(α)nY_n(α). SinceY_n(α)is bounded (actually it converges to 0), we have

X

x∈T∗

Cα_x <∞, MT−a.s..

This implies thata.s., for all but finitely many x,C_x <1, and thenC_x ≤C_xα, which gives the result. (2)As before, we haveα such that ρ(α) =inf0≤t≤1ρ(t)≤ 1. We have to distinguish two cases.

Eitherρ′(1)_≥0, therefore it is easy to see that, forα, (H2) is not verified, so

X

x∈Tn

C_xα=Y_n(α)→0,

when n goes to∞. Then fornlarge enough,C_x <1 for every x∈T_n, whence

X

x∈Tn

Cx→0,

then by themax-flow min-cut theorem, the associated capacited network admits no flow a.s., this implies that no electrical current flows, and that the random walk is recurrentMT-a.s..

We now deal with the case whereρ′(1)<0, thenα=1. The proof is similar to[16], but, as it is quite short, we give it for the sake of clarity. We use the fact that, if the capacited network admits no flow frome, then the walk is recurrent.

We call F the maximum flows fromein T, and for x∈T, |x|=1, we call Fx the maximum flow in the subtree T_x =_{y _∈T,x _≤ y_}, with capacity _AC_(xy₎ along the edge (←x−,x). It is easy to see that F

andF_x have the same distribution, and that

F= X

|x|=1

Taking the expectation yields

E[F] =E[Fx∧1] =E[F∧1], thereforeesssupF≤1. By independence, we obtain from (4) that

esssupF = (esssup X

|x|=1

A(x))(esssupF).

This implies thatF =0 almost surely, as(esssupP_|x|=1A(x))>1, whenP_|x|=1A(x)is not identi-cally equal to 1 .

(3)We shall use the fact that, if the water flows whenC_x is reduced exponentially in_|x_|, then the electrical current flows, and the random walk is transient a.s. (see[15]).

We have

inf

α∈[0,1]E

 N

(e)

X 0

A(e_i)α



=p>1 (pcan be infinite, in which case the proof still applies).

We introduce the measureµn defined as

µn(A) =E[♯(A∩ {logCx}x∈Tn)],

where♯denotes the cardinality. One can easily check that

φn(λ):=

Z +∞

−∞

eλtdµn(t) =E



X

x∈Tn

C_xλ



=ρ(λ)n. Let y ∈(0, 1]be such that p =inft>0y1−tE[

P

x∈T1A(x)

t_{]. Then, using Cramer-Chernov theorem} (and the fact that the probability measureµn/mn has the same Laplace transform as the sum ofn independent random variables with lawµ1/m), we have

1

nlogµn([n(−logy),∞))→log(p/y).

Now, if we set 1/y<q<p/y, there exists ksuch that

E[♯{x _∈T_k|C_x > yk_}]>qk.

Then the end of the proof is similar to the proof in[16]. We chose a smallε >0 such that,

E[♯{x ∈Tk|Cx > yk, and∀e<z≤x,A(z)> ε}]>qk.

Let Tk be the tree whose vertices are _{x _∈T_kn,n_∈N_{}such that x =}←−_{y in T}k _{iff x ≤ y in T and}

|y|=x+k. We form a random subgraphTk(ω)by deleting the edges(x,y)where

Y

x<z≤y

LetΓ₀ be the connected component of the root. The treeΓ₀ is a Galton-Watson tree, such that the expected number of children of a vertex isqk > 1, hence with a positive probabilityΓ₀ is infinite and has branching number overqk.

Using Kolmogoroff’s 0-1 Law, conditionally to the survival there is almost surely a infinite connected component, not necessarily containing the root. This connected component has branching number at leastqk. Then we can construct almost surely a subtree T′ ofT, with branching number overq, such that_∀x _∈T′,A(x)> εand if_|x_|=nk,_|y_|= (n+1)kandx < y thenQ_x<z≤yA(z)>qk. This implies the result.

We now turn to the proof of Proposition 1.1. Letπbe an invariant measure for the Markov chain (X_n,P_T)(that is a measure onT such that,_∀x_∈T,π(x) =P_y∈Tπ(y)ω(y,x)), then one can easily check that

π(x) = π(e)ω(e,

←−_e)

ω(x,←x−)

Y 0<z≤x

A(z), with the convention that a product over an empty set is equal to 1.

Then almost surely there exists a constantc>0 (dependant of the tree) such that

π(x)>c Cx.

Thus _X

x∈T

π(x)>cX

n

Y_n(1).

-Ifρ′(1)<0, then (H2) is verified andY >0 a.s. conditionally to the survival of the system, thus the invariant measure is infinite and the walk is null recurrent.

-Ifρ′(1) =0, we use a recent result from Y. Hu and Z. Shi. In[10] it was shown that, under the assumptions of Theorem 1.1, there exists a sequenceλn such that

0<lim inf n→∞

λn

n1/2 ≤lim sup_n→∞ λn

n1/2 <∞

andλnYn(1)→n→∞Y, withY >0 conditionally on the system’s survival. The result follows easily.

-If ρ′(1) > 0, there exists 0 < α < 1 such thatρ(α) = 1, ρ′(α) = 0. We set, for every x _∈ T, ˜

A(x):=A(x)α. We set accordingly ˜C(x) =Q_0<z≤xA˜(z), and

˜

ρ(t):=Eq

 N

(e)

X

i=1

˜

A(ei)t



=ρ(αt).

Note that ˜ρ(1) = 1 = inf0<t≤1ρ(t) and ˜ρ′(1) = 0. Note that under the ellipticity condition ω(x,←x−)> η, for some constantc>0

X

x∈T

π(x)<cX

x∈T

C_x =X

x∈T ˜

C1_x/α.

Using Theorem 1.6 of[10]withβ=1/αand ˜C_x =e−V(x), we get that for any 2₃α <r< α,

EMT

 



X

x∈Tn

Cx

 

r

Note that asr<1,

‚X

n

Y_n(1)

Œr

≤X

n

€

Y_n(1)Šr, whence, using Fatou’s Lemma,

EMT



 X

x∈T

Cx

!r

<∞. This finishes the proof.

3

The

_IMT

law.

We consider trees with a marked ray, which are composed of a semi infinite ray, calledRa y=_{v₀=

e,v₁ =←v−₀,v₂=←v−₁..._}such that to eachv_i is attached a tree. That wayv_i has several children, one of which beingvi−1.

As we did for usual trees, we can “mark” these trees with_{A(x)_}x∈T. Let ˜T_{be the set of such trees.} Let_Fn be the sigma algebraσ(Nx,Axi,vn ≤x) andF∞=σ(Fn,n≥0). While unspecified,

“mea-surable” will mean “F∞- measurable”.

Letqˆbe the law onN_×R∗N∗

+ defined by dˆq

dq =

N_X(e)

1

A(e_i).

Remark : For this definition to have any sense, it is fundamental that Eq[

PN(e)

1 Ai] =1, which is provided by the assumptionsρ′(1)<0 andp=1.

Following [21], let us introduce some laws on marked trees with a marked ray. Fix a vertex v0

(the root) and a semi infinite ray, calledRa y emanating from it. To each vertexv_∈Ra ywe attach independently a set of marked vertices with lawqˆ, except to the rooteto which we attach a set of children with law(q+ ˆq)/2. We chose one of these vertices, with probability _PA(vi)

A(vi), and identify it

with the child of v onRa y. Then we attach a tree with lawMTto the vertices not onRa y. We call

IMTthe law obtained.

We callθvT be the treeT “shifted” to v, that is,θvT has the same structure and labels as T, but its root is moved to vertexv.

Note that as before, given a treeT in ˜T_{, we can define in a unique way a familyω(x,y)such that}

ω(x,y) =0 unlessd(x,y) =1,

∀x _∈T,X y∈T

ω(x,y) =1, and

∀x_∈T,A(x) = ω(

←_x−_,x)

ω(←x−,←←−−x )

. (5)

We call random walk on T the Markov chain(Xt,PT) on T, starting from v0 and with transition

MT MT

MT MT MT MT MT

MT MT MT

v0

v1

v2

v3

h=1

h=0

h=₋1

h=₋2

h=₋3

h=₋4

(q+ ˆq)/2

ˆ q

Ray

Figure 2: TheIMTlaw. Let T_t = θXt_{T denote the walk seen from the particle. T}

t is clearly a Markov chain on ˜T. We set, for any probability measureµon ˜T_,P_µ=µ⊗P_{T the annealed law of the random walk in a random} environment on trees following the lawµ. We have the following

Lemma 3.1. IMTis a stationnary and reversible measure for the Markov process T_t, in the sense that,

for every F : ˜T2_→R_measurable,

E_IMT[F(T0,T1)] =EIMT[F(T1,T0)].

Proof : Suppose G is a_Fn-measurable function, that is, G only depends on the (classical) marked tree of the descendants of v_n, to which we will refer asT−nand on the position of v₀ in then₋th

level ofT−n. We shall write accordinglyG(T) =G(T−n,v0)

We first show the following

Lemma 3.2. If G isFn measurable, then

E_IMT[G(T)] =E_MT



X

x∈Tn

C_xG(T,x)

‚

1+PA(xi) 2

Œ

. (6)

Remark : These formulae seem to create a dependency on n, which is actually irrelevant, since

E_q[P_iN₌(₁e)A(e_i)] =1.

Proof : This can be seen by an induction overn, using the fact that

EIMT[G(T−n,v0)] =Eq

 XN

i=1

A(ei)E

”

G(T′(i,N,A(ej)),v0)|i,N,A(ej)

— ,

where T′(x,N,A(e_i))is a tree composed of a vertexv_n withN children marked with theA(e_i), and on each of this children is attached a tree with lawMT, except on the i-th, where we attach a tree whose law is the same asT−(n−1).

Iterating this argument we have

EIMT[G(T−n,v0)] =EMT



X

x∈Tn

CxE

G(T′′(x,T),x)_|x,T

 ,

where thenfirst levels of T′′(x,T) are similar to those of T, to each y _∈T_n′′, x ₆= y is attached a tree with lawMT, and to x is attached a set of children with law(ˆq+q)/2, upon which we attach

MTtrees. The result follows.

Let us go back to the proof of Lemma 3.1. Using the definition of the random walk, we get

E

IMT[F(T0,T1)] =EIMT

 X

x∈T

ω(v0,x)F(T,θxT)  .

Suppose F is_{F(n−2)× F(n−2)}measurable; then T →F(T,θxT)is at least F(n−1)measurable. Then

we can use (6) to get

E

IMT[F(T0,T1)] =EMT



X

x∈Tn

C_x

‚

1+PA(x_i) 2

ΠX

y∈T

ω(x,y)F(T,θyT)

 . It is easily verified that

∀x,y _∈T,ω(x,y)1+

P

A(x_i)

2 Cx =ω(y,x)

1+PA(y_i) 2 Cy. Using this equality, we get

E

IMT[F(T0,T1)] = EMT



X

x∈Tn X

y∈T

ω(y,x)C_y

‚

1+PA(y_i) 2

Œ

F((T,x),(T,y))

 

= E_MT

 

 X

y∈Tn+1

ω(y,←−y)C_y

‚

1+PA(y_i) 2

Œ

F((T,←−y),(T,y))

  

+ E_MT

 

 X

y∈Tn−1

X

i

ω(y,y_i)C_y

‚

1+PA(y_i) 2

Œ

F((T,y_i),(T,y))

  . Using (6) and the fact thatF is_{F(n−2)× F(n−2)}-measurable, we get

E

IMT[F(T0,T1)] = EIMT

h

ω(e,←−e)F(θ←−e T,T)i+E_IMT

 X

i

ω(e,e_i)F(θei_T,T)  

= E

IMT

F(T₁,T₀). This finishes the proof of (3.1).

4

The Central Limit Theorem for the RWRE on

_IMT

Trees.

In this section we introduce and show a central limit theorem for random walk on a tree following the law IMT. For T ∈ T˜, let h be the horocycle distance on T (see Figure 2). hcan be defined

recursively by ₍

h(v0) =0

h(←x−) =h(x)₋1, _∀x ∈T.

We have the following

Theorem 4.1. Suppose p = 1, ρ′(1) < 0and κ∈[5,_∞], as well as assumptions (2) and (H2) or

(3). There exists a deterministic constantσsuch that, forIMT₋a.e.T,the process_{h(X_⌊nt⌋)/pσ2_n}

converges in distribution to a standard Brownian motion, as n goes to infinity.

The proof of this result consists in the computation of a harmonic functionS_x on T. We will show that the martingaleS_X

t follows an invariance principle, and then thatSx stays very close toh(x).

Let, forv∈T,

W_v =lim

n

X

x∈T,v<x,d(v,x=n)

Y

v<z≤x

A(z).

Statement 2.1 implies thatW_v >0a.s. andE[W_v|σ(A(x_i),N(x),x < v)] =1. Now, let M₀=0 and ifX_t=v,

M_t+1−Mt=

¨

−Wv if Xt+1=←v−

W_v

i, ifXt+1=vi

.

GivenT, this is clearly a martingale with respect to the filtration associated to the walk. We intro-duce the functionSx defined asSe=0 and for allx ∈T,

Sxi =Sx+Wxi, (7)

in such a way thatMt=SXt.

Let

η=E_GW[W₀2], (8)

which is finite due to Theorem 2.1 of[13](the assumption needed for this to be true isκ >2). We call

V_t:= 1

t

t

X

i=1

E_T[(M_i+1−M i)2_|Ft] the normalized quadratic variation process associated toM_t. We get

E_T[(Mi+1−M i)2|Ft] =ω(Xi,←X−i)WX2i+

NX(Xi)

j=1

ω(Xi,Xi j)WX2_{i j}=G(Ti),

whereX_{i j} are the children ofXi andG is a L1(IMT)function on ˜T(again due toκ >2). Let us defineσsuch thatE_IMT[G(T)]:=σ2η2. We have the following

Proposition 4.1. The process {M⌊nt⌋/pσ2_η2_{n} converges, for IMT almost every T, to a standard}

Proof : We need the fact that whent goes to infinity,

V_t→σ2η2.

This comes from Birkhof’s Theorem, using the transformationθon ˜T_{, which conserves the measure}

IMT. The only point is to show that this transformation is ergodic, which follows from the fact that any invariant set must be independent of_Fnp=σ(N(x),A(x_i),v_n≤x,h(x)<p), for alln,p, hence is independent ofF_∞.

The result follows then from the Central Limit Theorem for martingales. Our aim is now to show thath(X_t) andM_t/ηstay close in some sense, then the central limit theorem forh(X_t)will follow easily.

Let

ε0<1/100,δ∈(1/2+1/3+4ε0, 1−4ε0)

and for every t, letρt be an integer valued random variable uniformly chosen in[t,t+⌊tδ⌋]. It is important to note that, by choosingε0 small enough, we can getδas close to 1 as we need.

We are going to show the following Proposition 4.2. For any0< ε < ε0,

lim

t→∞PT(|Mρt/η−h(Xρt)| ≥ε

p

t) =0, IMT₋a.s.,

further,

lim

t→∞PT _r,s<t,sup_|r−s|<tδ|

h(Xr)−h(Xs)|>t1/2−ε

!

=0, IMT−a.s..

Before proving this result, we need some notations. For any vertexv ofT, let

SRay

v =

X

yon the geodesic connectingvandRay,y6∈Ray

W_y.

We need a fundamental result on marked Galton-Watson trees. For a (classical) tree T, and x in T, set

S_x = X

e<y≤x

W_x, withWx as before, and

Aε_n=

¨

v∈T,d(v,e) =n,

S_v

n −η

> ε

«

. We have the following

Lemma 4.2. Let2< λ < κ−1,then for some constant C1 depending onε,

EMT

 

X

x∈Aε

n

Cx

 

Proof :We consider the setT∗_{of trees with a marked path from the root, that is, an element of}T∗ is of the form(T,v0,v1, ...), whereT is inT,v0=eandvi=←−−vi+1.

We consider the filtrationF_k=σ(T,v1, ...vk). Given an integern, we introduce the lawdMT∗nonT∗

de-fined as follows : we consider a vertexe(the root), to this vertex we attach a set of marked children with lawˆq, and we chose one of those children as v1, with probabilityP(x =v1) =A(x)/

P

A(ei). To each child ofedifferent fromv1we attach independently a tree with lawMT, and onv1we iterate

the process : we attach a set of children with lawqˆ, we choose one of these children to bev₂, and so on, until getting to the leveln. Then we attach a tree with lawMTtov_n.

ˆ q

MT MT MT

v0

v1

ˆ q

MT MT v2

vn−1

vn

MT MT MT MT

Figure 3: the lawMTd∗

n.

The same calculations as in the proof of Lemma 3.2 allow us to see the following fact : for f F_n-measurable,

E_dMT∗ n

[f(T,v0, ...,vn)] =EMT



X

x∈Tn

Cxf(T,p(x))



, (10)

where p(x) is the path from eto x. Note that, by construction, under MTd∗_n conditionally to ˜F_n∗ := (Cvi, 0≤i≤n), the trees T

(vi)_{, 0≤i≤nof the descendants of v}

i who are not descendants ofvi+1

are independent trees, and the law ofT(vi)_{is the law of a MTtree, except for the first level, whose}

law isˆqconditioned onvi+1,A(vi+1).

For a tree T inT∗_{we have}

W_v

k =

X

vk=←−x,x6=vk+1

A(x)W_x+A(v_k+1)W_v

k+1:=W

∗

k +A(vk+1)Wvk+1,

where

W_j∗= lim

n→∞

X

x∈T,vj<x,vj+16≤x,d(vj,x)=n Y

v≤z≤x

A(z). Iterating this, we obtain

W_vk =

n−1 X

j=k

W_j∗

j

Y

i=k+1

A(v_i) +W_vn

n

Y

i=k+1

with the convention that the product over an empty space is equal to one. We shall use the notation

Ai:=A(vi)for a tree with a marked ray. Finally, summing over k, we obtain

Svn=

n−1 X

j=0

W_j∗

j

X

k=0

j

Y

i=k+1

Ai+Wvn

n

X

k=0

n

Y

i=k+1

Ai. (11)

LetBj =

Pj k=0

Qj

i=k+1Ai. We note for simplicityWvn:=W

∗

n. Note that

E_dMT∗ n

[W0] =EMT

  



X

x∈Tn

Cx

 

2 

:=EMT[Mn2]

converges toη=E_MT[W₀2]asngoes to infinity. Indeed, recalling thatE_MT[M_n] =1, we have

EMT[(Mn+1−1)2] = Eq

  

 N

(e)

X

i=1

A(ei)Ui−1

 

2  

= Eq

  

 N

(e)

X

i=1

A(ei)(Ui−1) + N_X(e)

i=1

A(ei)−1

 

2  ,

where, conditionally to theAi, Ui are i.i.d. random variables, with the same law asMn. We get

E_MT[(M_n+1−1)2] =ρ(2)E_MT[(M_n−1)2] +C₂,

where C₂ is a finite number. It is easy to see then that E[M_n2]is bounded, and martingale theory implies that Mn converges in L2. Using the fact that EMTd∗_n[Wvk] =EØ_MT∗

n−k

[W0], a “Cesaro” argument

implies thatE_MTd∗ n

[S_vn]/nconverges toηasngoes to infinity. In view of that and (10) it is clear that, for n large enough

EMT

 

X

x∈Aε

n

Cx

 

 ≤ EMT



X

x∈Tn

Cx1Sx−EÔMT∗_n[Sx]>nε/2  

≤ P_d

MT∗_n

• S_v

n−EMTd∗_n[Svn|F˜

∗

n]

> nε

4

˜

+ P_d

MT∗_n

• E_d

MT∗_n[Svn|F˜

∗

n]−EMTd∗_n[Svn] > nε

4

˜

:=P₁+P₂. Let us first boundP1. Let ˜Wj∗:=Wj∗−EdMT∗_n[W

∗

j|F˜n∗]andλ∈(2,κ−1). We have

E(_n1) := E_MTd∗ n

Svn−EdMT∗_n[Svn|F˜

∗ n] λ

=E_MTd∗ n    n X

i=0

˜

W_i∗Bi

λ  

= E_MTd∗ n

  E_MTd∗

n    n X

i=0

˜

W_i∗Bi

λ

|F˜_n∗

     .

Inequality from page 82 of[22]implies

E_n(1)_≤C(λ)nλ/2−1E_d

MT∗_n

 Xn

i=0

E_d

MT∗_n

h€

˜

W_i∗B_iŠλ_|F˜_n∗i



_≤C₃nλ/2−1E_d

MT∗_n

 Xn

i=0

Bλ_i

 , where we have admitted the following lemma

Lemma 4.3. _∀µ < κ,there exist some constant C such that E_MT∗

n[(W

∗

i )

µ

|F˜_n∗]<C. (12)

moreover, there exists someǫ1>0such that

E_MT∗ n[W

∗

i |F˜n∗]> ǫ1.

We postpone the proof of this lemma and finish the proof of Lemma 4.2. In order to boundE_dMT∗ ”

Bλ_i—

we need to introduce a result from[4](lemma 4.1).

Statement 4.1(Biggins and Kyprianou). For any n_≥1and any measurable function G,

E_MT



X

x∈Tn

C_xG(C_y,e< y_≤x)



=E[G(eSi_{; 1≤i≤n)],}

where S_n is the sum of n i.i.d variables whose common distribution is determined by

E[g(S₁)] =E_q

 

N_X(e)

i=1

A(e_i)g(logA(e_i))

 

for any positive measurable function g.

In particular,E[eλS1_{] =E}

q[

PN(e)

i=1 A(ei)λ+1] =ρ(λ+1)<1. We are now able to compute

E_MTd∗ n

”

Bλ_n—=E_MT

 

X

x∈Tn

C_x



 X

e≤y≤x

Y

y<z≤x

A(z)

 

λ

 =E

  

n

X

k=0

eSn−Sk !λ

 . Using Minkowski’s Inequality, we get

E_MTd∗ n

”

Bλ_n—≤

n

X

k=0

E”eλ(Sk−Sn)—

1 λ !λ ≤ n X

k=0

ρ(λ+1)n−λk

!λ

≤C4. (13)

We can now conclude,

E_n(1)_≤C₅nλ/2, and by Markov’s Inequality,

P₁<C₆/(ελnλ/2). (14)

Now we are going to deal with

P₂=P_MTd∗ n

• E_MTd∗

n

[S_vn|F˜_n∗]₋E_MTd∗ n

[S_vn]

>nε/2

˜

Lemma 4.3 implies that E_d

MT∗_n[W

∗

j|F˜n∗] is bounded above and away from zero, and a deterministic function ofAj+1. We shall note accordingly

E_d

MT∗_n[W

∗

j|F˜n∗]:=g(Aj+1). (15)

Recalling (11), we have

E_MTd∗ n

[Svn|F˜

∗

n] = n

X

j=0

E_MTd∗ n

[W_j∗|F˜_n∗])Bj=

X 0≤j≤k≤n

k

Y

i=j

Aig(Ak+1).

with the conventiong(An+1) =1 andA0=1. We set accordingly

E_d

MT∗_n[Svn|F˜

∗

n]:=F(A1, ...,An). Recalling that, due to Statement 4.1, under the lawMTd∗

n, theAi are i.i.d random variables we get

E_dMT∗ n

[F(A1, ...,An)] =

X 0≤j≤k≤n

k

Y

i=j

E_MTd∗ n

[Ai]EMTd∗_n[g(Ak+1)].

Form_≥0 we call

Fm[A_m+1, ...,A_n]

:= X

0≤j≤k≤n k≤m−1

k

Y

i=j

E_MTd∗ n

[Ai]EdMT_n∗[g(Ak+1)] +

X 0≤j≤k≤n

k≥m m

Y

i=j

E_MTd∗ n

[Ai] k

Y

i′_=m+1

Ai′g(A_k+1).

Note thatF0=F andFn=E_d

MT∗_n[Svn], thus we can write

E_d

MT∗_n[Svn|F˜

∗

n]−EMTd∗_n[Svn] =F 0_(A

1, ...An)−Fn

=F0(A₁, ...A_n)₋F1(A₂, ...A_n) +F1(A2, ...An)−F2(A3, ...An)....

+Fn−1(A_n)₋Fn.

We introduce the notations ρ := E_MTd∗ n

[A1] = ρ(2) < 1, and for a random variable X, ˜X := X −

E_d

The last expression gives us

E_MTd∗ n

[Svn|F˜

∗

n]−EMTd∗_n[Svn]

=g˜(A₁) +A˜₁(g(A₂) +A₂g(A₃) +...+ n

Y

i=2

A_ig(A_n+1))

+ρg˜(A2) +A˜2(1+ρ)   

n

X

j=3

j

Y

i=3

Aig(Aj+1)   

+ρ2˜g(A3) +A˜3(1+ρ+ρ2)   

n

X

j=4

j

Y

i=4

Aig(Aj+1)   +... +ρn−1˜g(An) +A˜n(1+ρ+ρ2+...ρn−1).

We deduce easily that

E_MTd∗

n

[S_vn|F˜_n∗]₋E_dMT∗ n

[S_vn]

<C₇+C₈

n X

k=1

˜

A_kD_k(1+ρ+ρ2+...ρk−1)

, (16)

whereC7,C8 are finite constants and

Dk=

n

X

j=k+1

j

Y

i=k+1

Aig(Aj+1).

To finish the proof of Lemma 4.2, we need to show that for everyε >0, P_MTd∗ n

[Pn_k=1A˜_kDk(1+ρ+

ρ2+...ρk−1)>nε]< _nCλ/(ε)2−1.

Recalling thatλ < κ−1, we can find a smallν >0 such thatλ(1+ν)< κ−1. Then we have, by Minkowski’s Inequality

E_d

MT∗_n

h

Dλ(_k 1+ν)i_≤

  

n

X

j=k+1

E_d

MT∗_n

 C₈

n

Y

i=k+1

Aλ(_i 1+µ)

 

!1/λ(1+ν)

 

λ(1+ν)

≤    n X

j=k

€

C₉ρ(1+λ(1+µ))n−k+1Š1/λ(1+ν)

  

λ(1+ν)

<C₁₀.

(17) Markov’s Inequality then implies

P_MTd∗ n

max

k≤n Dn>(ε

2_n)2(11+ν)

≤C11

n

On the other hand, we call for 0_≤k_≤n,

N_k:=

n

X

j=n−k

D_jg(A_n+1)(1+ρ+ρ2+...ρj−1).

It is easy to check that N_k is a martingale with respect to the filtration_Hk=σ(A_j,n₋k_≤ j _≤n). We can compute the quadratic variation of this martingale

〈N_k〉:= k

X

j=1

E_d

MT∗_n[(Nk−Nk−1)

2

|Hk−1] =ρ(3)

k

X

j=1

(D_n−j)2.

On the other hand, the total quadratic variation ofNk is equal to

[N_k]:= k

X

j=1

(N_k−N_k−1)2= k

X

j=1

(A˜_n−jD_n−j)2.

It is easy to check that if the event in (18) is fullfilled, then there exists some constantC12such that

〈N_k〉<C12n 1+ 1

2(1+ν) _and[N

k]<C12n 1+ 1

2(1+ν)_{. Therefore, using (18) and Theorem 2.1 of[2],}

P_MTd∗ n

[_| n

X

k=1

˜

AkDk|>nε]≤C11

n

nλ/2ελ +2 exp−

(εn)2 2C₁₂n1+2(11+ν)

. (19)

Putting together (14) and (19), we obtain (9). This finishes the proof of Lemma 4.2. In particular, ifκ >5, we can chooseλ >4, so that

E_MT[X

x∈Aε

n

C_x]<n−µ,

withµ >1 . The following corollary is a direct consequence of the proof. Corollary 4.4. For every a>0and2< λ < κ−1,

P_d

MT∗_n[|Svk−kη|>a]≤C1

k1−λ/2

aλ .

We now give the proof of Lemma 4.3. As we said in the introduction, for this lemma we need either the assumption (H2) or the assumption (3). We give the proof in both cases. Note that, by construction of MT∗, as, using Theorem 2.1 of [13], for every x a child of v_i, different from v_i+1,

W(x)has finite moments of orderµ,

E_MT∗ n[(W

∗

i )

µ

|F˜_n∗] = C₀E_MT∗ n

  

 

 X

←−_x=v

i,x6=vi+1 A(x)

  

µ

_F˜∗

n

 

 (20)

= C0Eqˆ

  

 

 X

|x|=1,x6=v1 A(x)

  

µ

|A(v1)  

Note that the upper bound is trivial under assumption (3). We suppose (H2), Let f be a measurable test function, we have by construction

E_ˆq

  

 

 X

|x|=1,x6=v1 A(x)

  

µ

f(A(v₁))

  

= E_q

  

N(e)

X

i=1

A(e_i)

 

X

i6=j

A(e_j)

  

µ

f(A(e_i))

  

≤

∞ X

n=1

P_q(N(e) =n)E_q

  

n

X

i=1

A′(i)

 

X

i6=j

A′(1)

  

µ

f(A′(1))

  

By standard convexity property, we get that the last term is lesser or equal to

∞ X

n=1

Pq(N(e) =n)Eq

  

n

X

i=1

A′(i)nµ−1X

i6=j

A′(j)µf(A′(i))

  

≤E_q[A′(1)µ] ∞ X

n=1

P_q(N(e) =n)nµ+1E_qA′(i)f(A′(i)) =E_q[A′(1)µ]E_qA′(i)f(A′(i))E_q”N(e)µ+1—, while, still by construction

Eqˆ

f(A(v1))

= E_q

 

N(e)

X

i=1

A(e_i)f(A(e_i))

 

= ∞ X

n=1

Pq(N(e) =n)Eq[ n

X

i=1

A′(i)f(A′(i))] = E_q[N(e)]E_q[A′(1)f(A′(1))].

Therefore the result is direct. To prove the lower bound we begin with assumption (3). Actually we will only use the second part of this assumption, which is trivially implied by (H2), so the proof will also work for this case.

We have

Eˆq

 

 X

|x|=1,x6=v1

A(x)f(A(v₁))

  =E_q

  

N(e)

X

i=1

A(e_i)

 

X

i6=j

A(e_j)

 

f(A(e_i))

  

≥ ε0 ∞ X

i=1

Eq[A(ei)f(A(ei))1{i≤N(e)}(N(e)−1)]

≥ ε0 ∞ X

i=2

E_q[A(e_i)f(A(e_i))_1{i≤N(e)}(N(e)₋1)] +ε0Eq[A(e1)f(A(e1))(N(e)−1)]

≥ ε0 ∞ X

i=2

E_q[A(e_i)f(A(e_i))_1{i≤N(e)}] +ε0Eq[A(e1)f(A(e1))P(N(e)>2|A(e1))]

≥ ε0

N₀Eq

 

N_X(e)

i=1

A(ei)f(A(ei)

 =Eˆq

f(A(v1))

,

indeed fori_≥2, the event_{i<N(e)_}impliesN(e)₋1>1. This finishes the proof of Lemma 4.3. Let us go back toIMTtrees. We consider the following sets

Bε_n=

(

v_∈T,d(v,Ray) =n,

SRay_v

n −η

> ε

)

. (22)

We can now prove the following Lemma 4.5.

lim

t→∞PT(Xρt∈ ∪

∞

n=1B

ε

n) =0,IMT−a.s..

Proof : we recall that aIMTtree is composed of a semi-infinite path from the root : Ray=_{v0 =

e,v₁=←v−₀..._}, and that

W_j∗=lim

n

X

x∈T,vj<x,vj−16≤x,d(vj,x=n) Y

v≤z≤x

A(z).

Recalling Lemma 4.3, under IMT, conditionally to _{Ray,A(v_i)_}, W_j∗ are independent random vari-ables andE[W_j∗]> ǫ0.

Let 1/2< γ < δ. For a given tree T, we consider the event Γ_t =_{∃u_≤2t_|X_u=v_⌊tγ_⌋}. We have

Γ_{t⊂ {}inf

u≤2tMu≤Sv⌊tγ⌋},

andIMTalmost surely, for someε,

S_v

⌊tγ⌋≤ − ⌊tγ_⌋

X 0

Using Inequality 2.6.20 from page 82 of[23], we obtain that, for some constantC₂ E

S_vn−E_dMT∗

n

[S_vn|F˜_n∗] λ

<C₂

n X k=0

E[B_kλ].

Then, using the same arguments as in the proof of Proposition 4.2, we get that E[B_kλ]is bounded independently ofnandk, whence

E

S_vn−E_dMT∗

n

[S_vn|F˜_n∗] λ

<C₃n. Using Markov’s Inequality, there existsC₄ such that

P₁< C4

2 n

−(λ−1)_{. (48)}

On the other hand, recalling (16),

E_MTd∗

n

[Svn|F˜

∗

n]−EdMT∗_n[Svn]

<C5+

n X k=1

˜

AkDkg(An+1)(1+ρ+ρ2+...ρk−1) , whereC₅is a finite constant and

D_k= n X j=k+1

j Y i=k+1

A_ig(A_j+1),

where gis a bounded function. We recall that

N_k:= n X j=n−k

˜

A_jD_j(1+ρ+ρ2+...ρj−1)

is a martingale with respect to the filtration_Hk=σ(A_j,n₋k_≤ j_≤n), whence, using Burkholder’s Inequality,

E_d

MT∗_n[(Nn)

λ_]

≤C₆E_d MT∗_n

  

n X i=0

(D_i)2

!λ/2

 .

We recall that 1< λ <(κ−1)_∧2, whence, by concavity, the last expression is bounded above by

C₆E_MTd∗

n

 Xn

i=0 (D_i)λ

 <C₇n.

Therefore, using Markov’s Inequality, we get that

P2<n1−λ. This, together with (48), finishes the proof of Lemma 7.2.

We now finish the proof of Lemma 7.1. Let us go back toIMTtrees. We recall the definition of the setsBε_n:

Bε_n= (

v∈T,d(v,Ray) =n,

SRayv

n −η

> ε ) . (49)

We are going to prove that lim

t→∞PT(Xρt ∈ ∪

∞

n=1B ε

m) =0,IMT−a.s.. We introduceγ >1/2, and recall the definition of the event

Γ_t =_{∃u≤2t|X_u=v_⌊tγ_⌋}.

It is easy to see, using the same arguments as in the proof of Lemma 4.5, that

P_IMT[Γ_t] _→ t→∞0.

Furthermore, we introduce the event

Γ′_t=_{∃0_≤u_≤t,d(X_u,Ray)>nγ_}; then it is a direct consequence of Lemma 4.6 that

P_IMT[Γ′_t] _→ t→∞0.

As for the quenched case, we have

P

IMT(Xρt ∈ ∪

∞

m=1B ε

m) ≤ PIMT(Xρt ∈ ∪ nγ m=1B

ε m;Γ

c t∩Γ′

c

t) +PIMT(Γt) +PIMT(Γ′t)

≤ 1

⌊tδ_⌋EIMT   E_T

  

H_Xv_⌊tγ_⌋

s=0

1Xs∈∪t

γ

m=1Bεm   

 

+o(1), (50)

whereHv_⌊tγ⌋is the first time the walk hitsv⌊tγ⌋.

We recall thatT(vi)_{the subtree constituted of the verticesx ∈Tsuch thatv}

i≤x,vi−16≤ x, the same computations as in the proof of Lemma 4.5 imply

P

IMT(Xρt ∈ ∪

∞

m=1B ε m)≤

1

⌊tδ_⌋EIMT

  

⌊tγ⌋

X i=0

E_T

  

H_Xv_⌊tγ_⌋

s=0

1Xs=vi   _N˜_i

 

, (51)

Where ˜Ni is thePT−expectation of the number of visits to∪n

δ

m=1B ε m∩T

(vi)_{during one excursion in}

T(vi)_{. Lemma 7.2, and the method of 4.3 imply that, underIMTconditioned on{Ray,A(v}

i)}, ˜Ni are independent and identically distributed variables, with expectation at most equal to C₁′Pn_i=γ₀i1−λ

for someλ >1. By choosingγclose enough to 0, we get E_IMT[N˜_i|{Ray,A(v_i)_}]_≤C₁′n1/2−ǫfor some

ǫ >0. We recall that

E_T

  

H_Xv_⌊tγ_⌋

s=0

1Xs=vi   ≤C′′

 

1+

⌊t_Xγ⌋−1 j=0

⌊tγ⌋

Y k=j−1

A(vk)   .

The latter expression has bounded expectation underIMT, as an easy consequence of Statement 4.1 and Lemma 3.2.

We deduce that

P

IMT(Xρt ∈ ∪

∞

m=1B ε

m)≤C5n

1

2−ǫ+γ−δ.

Sinceγcan be chosen as close to 1/2 as needed, the exponent can be taken lower than 0. The end of the proof is then completely similar to the quenched case.

7.2

The annealed CLT on

MT

trees.

We now turn to the proof of Theorem 1.3. We use the coupling and the notations presented in section 5. Our main proposition in this part will be the following:

Proposition 7.2. Under the assumptions of Theorem 1.3, for someα <1/2 lim

t→∞PMT(∆t6= ∆

α

t) =0, (52)

and

lim

t→∞PIMT(∆˜t6=∆˜

α

t) =0. (53)

Further, underMT,

lim sup∆t

t =0, (54)

and underIMT,

lim sup∆˜t

t =0. (55)

Finally, underIMT,

lim sup_pBt

t =0. (56)

(Herelim supdenotes the limit in law.)

Before proving the latter proposition, we introduce some technical estimates, whose proof will be postponed.

Lemma 7.3. For allε >0 lim t→∞

P

MT

  

t1/2+ε

X i=1

(ηi−τi)<t  

=0, (57)

lim t→∞

P

IMT

  

t1/2+ε

X i=1

(η˜i−τ˜i)<t  

=0, (58)

lim

t→∞PMT(∃k≤It,Θi−1,Θi ∈Jk,|YΘi|>|YΘi−1|) =0, (59)

lim t→∞

P

IMT(∃k≤It, ˜Θi−1, ˜Θi∈J˜k,d(Y˜Θ˜i,Ra y)>d(Y˜Θ˜i−1,Ra y) =0, (60)

lim

t→∞PMT(Xs∈ ∪

tα

k=tα_−(logt)2A

ε

t for some s≤t}=0, (61)

lim t→∞

P

IMT(Xs∈ ∪t

α

k=tα_−(logt)2B

ε

k for some s≤t) =0. (62)

lim t→∞

P

MT

€

W_X

s>t

1/4−ǫ_{for some0}

≤s_≤tŠ=0 (63)

lim t→∞PIMT

€

W_Xs>t1/4−ǫfor some0_≤s_≤tŠ=0 (64)

We now turn to the proof of (52). As a consequence of (57) and (59) that, with P_MT probability approaching 1 asngoes to infinity,

t(Θ_2t1/2+ε)>t,

whence, using Lemmas 7.3 and 4.6, lim

t→∞PMT _s∈∪maxIt k=1Jk

|Y_{s| ≥}tα

!

≤lim sup t→∞

2t1/2+ε

X i=0

P

MT ∃j>i:|YΘj| ≥t α

−(logt)2,Y_Θ

i =e,

S_YΘ

j ≥(η−ε1)t α

/2,_|YΘk|>0,∀i<k≤ j;WXs≤t 1/4−ǫ

∀0_≤s_≤t |SXs− |Xs|| ≤εt

1/4−ε′

|Xs|,∀s≤t

:=lim sup t→∞

2tX1/2+ε

i=1

P_i,t;

where ε,ε1 are positive numbers that can be chosen arbitrarily small. We recall that the process

{N_s}=_{S_X

θi+s∧Kt}is a supermartingale. and that there exists a previsible and non-decreasing process

A_s such that N_s+A_s is a martingale. Furthermore, on the event _{W_{Xs ≤} t1/4−ǫ_∀0 _≤ s _≤ t_}, the increments of this martingale are bounded above by t1/2−ǫ. Azuma’s Inequality implies the result, as in the quenched case.

The proof of (53) is similar and omitted.

We recall that in the proofs of (30),(31) and (32) we only used the assumptionκ >5, therefore the proof of (54),(55) and (56) are direct consequence, by dominated convergence.

We now turn to the proof of Lemma 7.3. The proofs of (57), (58), (59), (60) and (61) follow directly from equations (33), (34), (37), (38) and (39), whose proofs did not use any assumption other thanκ >5, by dominated convergence.

To prove (62), note that, similarly to the proof of 7.1,

P_IMT(X_{s∈ ∪}tα

k=tα−(logt)2B

ε

k for somes≤t)

=E_IMT

  

⌊tγ_⌋

X i=0

E_T

  

Hv

⌊tγ⌋

X s=0

1Xs=vi   N_i′

  ,

whereN_i′is theP_{T−expectation of the number of visits to∪}tα

k=tα−(logt)2B

ε k∩T

(vi)_{during one excursion} inT(vi)_{. Lemma 4.2 and the method of Lemma 4.3 imply that, underIMTconditioned on{Ray,A(v}

i)},

N_i′are independent and identically distributed variables, up to a bounded constant, with expection at most equal toC′(logt)2t−α(λ−1)for someλ >2. We also recall that

E_T

  

Hv

⌊tγ⌋

X s=0

1Xs=vi   ≤C′′

 

1+

⌊tγ_⌋−1

X j=0

⌊tγ_⌋

Y k=j−1

A(v_k)   .

has bounded expectation underIMT, as an easy consequence of Statement 4.1 and Lemma 3.2. By choosingγclose enough to 0 andαclose to 1, we get the result.

The proofs of (63) and (64) are easily deduced from the proofs of (35) and (36), the only difference being that we do not need to apply the Borel-Cantelli Lemma.

Acknowledgement:We would like to warmly thank Y. Hu for his help and encouragement.

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