Einstein relation for biased random walk on Galton-Watson trees

www.imstat.org/aihp 2013, Vol. 49, No. 3, 698–721

DOI:10.1214/12-AIHP486

© Association des Publications de l’Institut Henri Poincaré, 2013

Einstein relation for biased random walk on Galton–Watson trees

Gerard Ben Arous

, Yueyun Hu

, Stefano Olla

and Ofer Zeitouni

aCourant Institute, New York University, 251 Mercer St., New York, NY 10012, USA. E-mail:benarous@cims.nyu.edu bDépartement de Mathématiques, LAGA, Université Paris 13, 99 Av. J-B Clément, 93430 Villetaneuse, France.

E-mail:yueyun@math.univ-paris13.fr

cCEREMADE, Place du Maréchal de Lattre de TASSIGNY, F-75775 Paris Cedex 16, France and INRIA, Projet MICMAC, Ecole des Ponts, 6 & 8 Av. Pascal, 77455 Marne-la-Vallée Cedex 2, France. E-mail:olla@ceremade.dauphine.fr dSchool of Mathematics, University of Minnesota, 206 Church St. SE, Minneapolis, MN 55455, USA and Faculty of Mathematics,

Weizmann Institute, POB 26, Rehovot 76100, Israel. E-mail:zeitouni@math.umn.edu Received 25 June 2011; revised 26 February 2012; accepted 5 March 2012

Abstract. We prove the Einstein relation, relating the velocity under a small perturbation to the diffusivity in equilibrium, for certain biased random walks on Galton–Watson trees. This provides the first example where the Einstein relation is proved for motion in random media with arbitrarily slow traps.

Résumé. Nous prouvons la relation d’Einstein pour certaines marches aléatoires biaisées sur des arbres de Galton–Watson. Cette formule relie la dérivée de la vitesse à la diffusivité à l’équilibre. Ce travail fournit le premier exemple de preuve de la relation d’Einstein pour une dynamique dans un milieu aléatoire qui comporte des pièges arbitrairement lents.

MSC:Primary 60K37; secondary 60J80; 82C44

Keywords:Galton–Watson tree; Einstein relation; Spine representation

1. Introduction

Letωbe a rooted Galton–Watson tree with offspring distribution{pk}, wherep0=0,m=

kpk>1 and b^kpk<

∞for someb >1. For a vertexv∈ω, let|v|denote the distance ofvfrom the root ofω. Consider a (continuous-time) nearest-neighbor random walk{Y_t^α}t≥0onω, which when at a vertexv, jumps with rate 1 toward each child ofvand at rateλ=λ_α=me^−α,α∈R, toward the parent ofv.

It follows from [15] that ifα=0, the random walk{Y_t^α}t≥0is, for almost every treeω, null recurrent (positive re- current forα <0, transient forα >0). Further, an easy adaptation of [19] shows that|Y_[⁰_nt]|/√

nsatisfies a (quenched, and hence also annealed) invariance principle (i.e., converges weakly to a multiple of the absolute value of a Brownian motion), with diffusivity

D⁰= 2m²(m−1)

k²p_k−m. (1.1)

1Supported in part by NSF grant OISE-07-30136.

2Supported in part by the European Advanced GrantMacroscopic Laws and Dynamical Systems(MALADY) (ERC AdG 246953) and by grant ANR-2010-BLAN-0108 (SHEPI).

3Supported in part by NSF grant DMS-0804133 and by a grant from the Israel Science Foundation and the Taubman professorial chair at the Weizmann Institute.

(Compare with [19], Corollary 1, and note that the factor 2 is due to the speed up of the continuous-time walk relative to the discrete-time walk considered there. See (2.10) below and also the derivation in [5].) On the other hand, see [17], when α >0, |Y_t^α|/t→t→∞v¯_α>0, almost surely, withv¯_α deterministic. A consequence of our main result, Theorem1.2below, is the following.

Theorem 1.1 (Einstein relation). With notation and assumptions as above,

αlim0

¯ v_α

α =D⁰

2 . (1.2)

The relation (1.2) is known as anEinstein relation. It is straight forward to verify that for homogeneous random walks onZ+ (corresponding to deterministic Galton–Watson trees, that is, those withpk=1 for somek≥1), the Einstein relation holds.

In a weak limit (velocity rescaled with time) the Einstein relation is proved in a very general setup by Lebowitz and Rost (cf. [12]). See also [4] for general fluctuation-dissipation relations.

For the tagged particle in the symmetric exclusion process, the Einstein relation has been proved by Loulakis in d≥3 [13]. The approach of [13], based on perturbation theory and transient estimates, was adapted for bond diffusion inZ^dfor special environment distributions (cf. [10]). For mixing dynamical random environments with spectral gap, a full perturbation expansion can also be proved (cf. [11]).

For a diffusion in random potential, the recent [9] proves the Einstein relation by following the strategy of [12], adding to it a good control (uniform in the environment) of suitably defined regeneration times in the transient regime.

A major difference in our setup is the possibility of having “traps” of arbitrary strength in the environment; in partic- ular, the presence of such traps does not allow one to obtain estimates on regeneration times that are uniform in the environment, and we have been unable to obtain sharp enough estimates on regeneration times that would allow us to mimic the strategy in [9]. On the other hand, the tree structure allows us to develop some estimates directly for hitting times via recursions, see Section3. We emphasize that our work is (to the best of our knowledge) the first in which an Einstein relation is rigorously proved for motion in random environments with arbitrary strong traps.

In order to explore the full range of parameters α, we will work in a more general context than that described above, following [19]. This is described next.

Consider infinite trees T with no leaves, equipped with one (semi)-infinite directed path, denotedRay, starting from a distinguished vertex called therootand denotedo. We call such a tree amarked tree. UsingRay, we define in a natural way the offsprings of a vertexv∈T, and denote byD_n(v)the collection of vertices that are descendants of v at distancenfromv, withZ_n(v)= |D_n(v)|. See [19], Section 4, for precise definitions. For any vertexv∈T, we letd_vdenote the number of offspring ofv, and write^←v for the parent ofv. Finally, we writeρ(v)for thehorocycle distance ofvfrom the rooto. Note thatρ(v)is positive ifvis a a descendant ofoand negative if it is an ancestor ofo.

LetΩ_T denote the space of marked trees. As in [19] and motivated by [16], given an offspring distribution{p_k}k≥0

satisfying our general assumptions, we introduce a reference probabilityIGWonΩ_T, as follows. Fix the rootoand a semi-infinite ray, denoted Ray, emanating from it. Each vertexv∈Raywithv =o is assigned independently a size-biased number of offspring, that isPIGW(d_v=k)=kp_k/m, one of which is identified with the descendant ofvon Ray. To each offspring ofv =onot onRay, and too, one attaches an independent Galton–Watson tree of offspring distribution{p_k}k≥0. Note thatIGWmakes the collection{d_v}v∈T independent. We denote expectations with respect toIGWby·0(the reason for the notation will become apparent in Section2.1below).

As mentioned above and in contrast with [16], [17] and [19], it will be convenient to work in continuous time, because it slightly simplifies the formulas (the adaptation needed to transfer the results to the discrete time setup of [17] are straight-forward). For background, we refer to [5], where the results in [17] and [19] are transferred to continuous time, in the more general setup of multi-type Galton–Watson trees. Given a marked treeT andα∈R, we define anα-biased random walk{X_t^α}t≥0onT as the continuous time Markov process with state space the vertices of T,X^α₀=o, and so that when atv, the jump rate is 1 toward each of the descendants ofv, and the jump rate is e^−αm toward the parent^←v. More explicitly, the generator of the random walk{X^α_t}t≥0can be written as

Lα,TF (v)=

x∈D1(v)

F (x)−F (v)

+e^−αm

F (^←v )−F (v)

. (1.3)

Alternatively,

Lα,TF (v)= −m∂₋^∗∂₋F (v)+

e^−α−1

m∂₋F (v), (1.4)

where∂₋F (v)=F (^←v )−F (v)and

∂₋^∗F (v)= 1 m

x∈D₁(v)

F (x)−F (v). (1.5)

Note that ifα <0 the (average) drift is towards the ancestors, whereas ifα >0 the (average) drift is towards the children. As in [17] and [19], we have that

tlim→∞

ρ(X_t^α)

t →t→∞vα, IGW-a.s. (1.6)

It is easy to verify that whenα >0, thenv_α = ¯v_α, and that sign(vα)=sign(α). Further, we have, again from [19], thatρ(X⁰_[nt])/√

nsatisfies the invariance principle (that is, converges weakly to a Brownian motion), with diffusivity constantD⁰as in (1.1).

Our main result concerning walks on IGW-trees is the following.

Theorem 1.2. With assumptions as above,

αlim→0

v_α α =D⁰

2 . (1.7)

Remark 1.3. It is natural to expect that the Einstein relation holds in many related models,including Galton–Watson trees with only moment bounds on the offspring distribution,multi-type Galton–Watson trees as in[5],and walks in random environments on Galton–Watson trees,at least in the regime where a CLT with non-zero variance holds,see [6].We do not explore these extensions here.

The structure of the paper is as follows. In the next section, we consider the case ofα <0, exhibit an invariant measure for the environment viewed from the point of view of the particle, and use it to prove the Einstein relation when α0. Section3 deals with the harder case of α0. We first prove an easier Einstein relation (or linear response) concerning escape probabilities of the walk, exploiting the tree structure to introduce certain recursions.

Using that, we relate the Einstein relation for velocities to estimates on hitting times. A crucial role in obtaining these estimates, and an alternative formula for the velocity (Theorem3.7), is obtained by the introduction, after [7], of a spine random walk, see Lemma3.3.

Remark 1.4. After this paper was completed,in an impressive work,Aïdékon[1]obtained a representation of the invariant measure of the environment viewed from the point of view of the particle,for the related discrete time model corresponding toα >0.This allowed him to give,as his main result,a representation of the velocity[1],Theorem1.1, which is different from that in Theorem3.7and avoids the use of the spine random walk;Aïdékon’s formula is also useful in studying properties of the velocity away from the pointα=0,for positiveα.Together with Proposition3.1, his formula could be used,forα0,to give an alternative proof of the hard half of Theorem1.1,without introducing the spine random walk.

2. The environment process, and the proof of Theorem1.2forα0

As is often the case when motion in random media is concerned, it is advantageous to consider the evolution, inΩ_T, of the environment from the point of view of the particle. One of the reasons for our opting to work in continuous time is that whenα=0, the invariant measure for that (Markov) process is simplyIGW, in contrast with the more complicated measureIGWRof [19]. We will see that whenα <0, an explicit invariant measure for the environment viewed from the point of view of the particle exists, and is absolutely continuous with respect toIGW.

2.1. The environment process

For a given treeT andx∈T, letτ_x denote the shift that moves the root ofT tox, withRayshifted to start atx in the unique way so that it differs fromRaybefore the shift by only finitely many vertices. Thenτ_xT is rooted atxand has the same (nonoriented) edges asT. (A special role will be played byτ_xforx∈D1(o), and byτ_xwithx=^←o. We useτ⁻¹T =τ←

oT in the latter case.) The environment process{Tt}t≥0is defined byTt=τ_XtT. It is straightforward to check that the environment process is a Markov process. In fact, introducing the operators

Df (T)=f τ⁻¹T

−f (T),

we have that the adjoint operator (with respect toIGW) is D^∗f (T)= 1

m

x∈D1(o)

f (τ_xT)−f (T)

since

gDf0= f D^∗g

0. Notice thatD^∗1=d_o/m−1.

DefineW (v, n)=Zn(v)/mⁿ. ThenW (v, n)is a positive martingale that converges to a random variable denoted W_v. Using the recursions

mW_v=

x∈D₁(v)

W_x, mW (v, n)=

x∈D₁(v)

W (n−1, x),

we see that W_v0=1 for v /∈Ray. To simplify notation, we writeW_−j =W_vj withv_j ∈Raydenoting the jth ancestor ofo. SinceW_o(τ_xT)=W_x(T), we have thatD^∗W_o=0.

The generator of the environment process is Lαf (T)=

x∈D1(o)

f (τxT)−f (T) +e^−αm f

τ⁻¹T

−f (T)

= −mD^∗Df (T)+

e^−α−1

mDf (T). (2.1)

The adjoint operator (with respect toIGW) isL^∗_α = −mD^∗D+(e^−α−1)mD^∗. For anyα∈R, letμ_α denote any stationary probability measure forL_α, that isμ_α satisfies, for any bounded measurablef,

L_αfα=0, wheregα=

gdμα.

Note that IGWis stationary andreversibleforL₀. Further, it is ergodic for the environment process. This is el- ementary to prove, since for any bounded functionf (T)such thatL0f =0, we have that|Df|²0=0, i.e.f is translation invariant for a.e.T with respect to IGW, i.e. constant a.e. Thus, necessarily, μ₀=IGW, justifying our notation·0= ·IGW.

In our setup, due to the existence of regeneration times for α =0 with bounded expectation, a general ergodic argument ensures the existence of a stationary measure μ_α, which however may fail in general to be absolutely continuous with respect toIGW, see [17]. Further, becauseIGWis ergodic and the random walk is elliptic, there is at most oneμ_α which is absolutely continuous with respect toIGW, since under any suchμ_α, the processTt^α must be ergodic, see e.g. [20], Corollary 2.1.25, for a similar argument. As we now show, whenα <0, this stationary measure μαwith density with respect toIGWcan be constructed explicitly.

Lemma 2.1. Forα <0,the probability measureμ_α=ψ_αμ₀where

ψα(T)=C_α⁻¹Zα, (2.2)

Z_α=^∞

j=0

e^{j α}W_−j(T),

(2.3) C_α=(1−b)e^αm⁻¹

1−e^αm⁻¹ + be^α

1−e^α +1, b=

kk²p_k−m m(m−1) , is stationary forLα.Furthermore

αlim0ψ_α(T)=1 μ₀-a.e. (2.4)

Proof. We show first thatC_α provides the correct normalization. In fact, from the relation W₋j=m⁻¹W₋j+1+m⁻¹

s∈D1(v₋j),s /∈Ray

Ws=:m⁻¹(W₋j+1+Lj) (2.5)

and sinceW_s0=1 ifs /∈Ray, we obtain

W₋j0=m⁻¹W₋j+10+m⁻¹b(m−1), j≥1. (2.6)

SinceW_o0=1, we deduce that

W_−j0=(1−b)m^−j+b, j≥0. (2.7)

Thus, _∞

j=0

e^{j α}W₋j

0

=1+b ∞ j=1

e^{j α}+(1−b) ∞ j=1

e^{j α}m^−j

=1+ be^α

1−e^α +(1−b)e^αm⁻¹ 1−e^αm⁻¹

= b

1−e^α + 1−b

1−e^αm⁻¹=C_α as needed.

Note that the termsL_j appearing in the right side of (2.5) are i.i.d. Substituting and iterating, we get W_−k=W_o

m^k +L₁ m^k + L₂

m^k−1 + · · · +L_k m. Therefore,

1−e^α

m

Z_α=W_o+ 1 m

∞ j=1

e^αjL_j=:W_o+M_α. (2.8)

Note thatM_αis a weighted sum of i.i.d. random variables. Further, because|D₁(v_−j)|0=

k²p_k/mandW_s0=1, we have that limα0|α|M_α0=(

k²p_k−m)/m²:= ¯C, and that VarIGW(M_α)=O(_|α¹_|). It then follows (by an interpolation argument) that that

αlim0|α|M_α= ¯C, IGW-a.s.

Substituting in (2.8), this yields

αlim0αZ_α=b IGW-a.s.

and (2.4) follows.

We next verify thatL^∗_αψ_α=0. SinceW_−j(τ⁻¹T)=W_−j−1(T), we have Dψ_α =C_α⁻¹

∞ j=0

e^{j α}

W_−j−1(T)−W_−j(T)

=C_α⁻¹ _∞

j=1

e^(j−1)αW_−j(T)−^∞

j=0

e^{j α}W_−j(T)

=C_α⁻¹ ∞ j=0

e^(j−1)α−e^{j α}

W₋j(T)−C_α⁻¹e^−αWo

=

e^−α−1 C_α⁻¹

∞ j=0

e^{j α}W₋j(T)−C_α⁻¹e^−αWo

=

e^−α−1

ψ_α−C_α⁻¹e^−αW_o. SinceD^∗Wo=0, we have

D^∗

Dψα−

e^−α−1 ψα

=0,

i.e.

L^∗_αψ_α=0.

We can now provide the proof of Theorem1.2in caseα0.

Proof of Theorem1.2(Caseα0). We begin with the computation ofv_α. Becauseμ_α is ergodic and absolutely continuous with respect toIGW, we have thatv_α equals the average drift (underμ_α) ato, that is

vα=m d_o

m −e^−α

α

=m D^∗1

α−m

e^−α−1

=m D^∗1ψα

0−m

e^−α−1

=mDψ_α0−m

e^−α−1

=m

e^−α−1

ψ_α−C_α⁻¹e^−αW_o

0−m

e^−α−1

= −mC_α⁻¹e^−α. Thus,

αlim0

v_α

|α|= − m²(m−1)

k²p_k−m. (2.9)

It remains to compute the diffusivityD⁰whenα=0. Toward this end, one simply repeats the computation in [19], Corollary 1. One obtains that the diffusivity is

D⁰=mW_o²+

s∈D₁(o)W_s²0

W_o²²₀ . (2.10)

From the definitions we have thatW_o²0=(

k²p_k−m)/m(m−1)(see [19], (2)), and thus D⁰= 2m²(m−1)

k²p_k−m. (2.11)

Together with (2.9), this completes the proof of Theorem1.2whenα0.

Remark 2.2. Note that the construction above fails forα >0,because thenZ_α is not defined.The caseα= ∞is however special.In that case,the generator is

L_∞f (T)=

x∈D1(o)

f (τxT)−f (T). (2.12)

In particular,one can verify that the measure defined bydμ_∞/dμGW=1/(Cdo)withC=

kk⁻¹pk andμGW the ordinary Galton–Watson measureGW(defined asIGWbut with the standard Galton–Watson measure also for vertices onRay),is a stationary measure,and thatv_∞= do∞=C.It follows that the natural invariant measure is not absolutely continuous with respect toIGW.

Remark 2.3. Forα <0one can construct other invariant measures,that of course are singular with respect toIGW.

A particular family of such measures is absolutely continuous with respect to the ordinary Galton–Watson measure GW.Indeed,one can verify that the positive function

ψ (T)=C ∞ j=1

j−1

i=1

d_−i me^−α =C

∞ j=1

me^−α₋j+1 j−1

i=1

d₋i (2.13)

withC=1−e^α,satisfies

ψdGW=1andψdGWis an invariant measure forL_α.One can also check that the Einstein relation(1.2)is not satisfied under this measure,emphasizing the role that the measureIGWplays in our setup.

3. Drift towards descendants: The proof of Theorem1.2forα0

In the caseα >0 we cannot find an explicit expression for the stationary measure so we have to proceed in a different way. We first prove another form of the Einstein relation in terms of theescape probabilities (probability of never returning to the origin).

Because we consider the caseα >0, there is no difference between considering the walk under the Galton–Watson tree or underIGW– the limiting velocity is the same, i.e.vα= ¯vα. Thus, we only consider the walk{Y_t^α}t≥0below.

Our approach is to provide an alternative formula for the speedv_α, see Theorem 3.7below, which is valid for allα >0 small enough. In doing so, we will take advantage of certain recursions, and of thespine random walk associated with the walk on the Galton–Watson tree, see Lemma3.3.

We recall our standing assumptions:p0=0,m >1, and

b^kpk<∞for someb >1. We will throughout drop the superscriptαfrom the notation when it is clear from the context, writing e.g.Y_t forY_t^α. To introduce our recursions, defineT (x):=inf{t≥0: Yt =x}andτn:=inf{t≥0: |Yt| =n}. For a given treeω, we writePx,ωfor the law ofYt

withY₀=x. For 0<|x| ≤n, define β_n(x):=P_x,ω

T (^←x ) > τ_n

, β(x):=P_x,ω

T (^←x )= ∞ , γ_n(x):=E_x,ω

τ_n∧T (^←x ) .

We study the recursions forβ_nandγ_n. By the Markov property ofP_x,ω, for|x|< n, γ_n(x)= 1

d_x+λ+

d_x

i=1

1

λ+d_xE_xi,ω

τ_n∧T (^←x )

= 1 d_x+λ+

dx

i=1

1 λ+d_x

Ex_i,ω

τn∧T (x) +Px_i,ω

T (x) < τn

Ex,ω

τn∧T (^←x ) ,

which implies that γ_n(x)= 1

d_x+λ+

d_x

i=1

1 λ+d_x

γ_n(x_i)+

1−β_n(x_i) γ_n(x)

.

Hence for any 0<|x|< n, γ_n(x)= 1+dx

i=1γn(xi) λ+dx

i=1βn(xi)

with boundary conditionγ_n(x)=0 for any|x| =n. We take the above equality as the definition ofγ_n(o). Similarly, we have

β_n(x)= dx

i=1β_n(x_i) λ+dx

i=1βn(xi)

, 0<|x|< n,

withβ_n(x)=1 if|x| =n, and we defineβ_n(o)so that the above equality holds forx=o. Finally, we letβ(o)= limn→∞β_n(o)(the limit of the monotone sequenceβ_n(o)).

Proposition 3.1. Asα0,α⁻¹β(o)converges in law and in expectation to a random variableY such that E(Y )= m(m−1)

E(d_o²−d_o)=D⁰

2m. (3.1)

This is a form of Einstein relation, as linear response for the escape probability. The law ofY can be identified, see the end of the proof of Proposition3.1.

Proof of Proposition3.1. We clearly have that withB(x):=¹_λd_x

i=1β(x_i), it holds that β(x)= B(x)

1+B(x) ∀x =o, and

B(x)=1 λ

dx

i=1

B(x_i)

1+B(x_i) ∀x∈T. (3.2)

Notice that allB(x)are distributed as some random variable, sayB, and conditionally ondx and on the tree up to generation|x|, the variablesB(xi),1≤i≤dxare i.i.d. and distributed asB. It follows that

E(B)=e^αE B

1+B (3.3)

and E

B²

= 1 λ²

mE

B 1+B

2 +E

do(do−1) E

B 1+B

2

. (3.4)

For any nonnegative r.v.Z∈L², let us denote by{Z} :=_E(Z)^Z . By concavity, the following inequality holds (see e.g.

[18], Lemma 6.4):

E Z

1+Z 2

≤E {Z}²

.

By (3.3) and (3.4), we get that E

{B}²

= 1 mE

B 1+B

2

+E(d_o(d_o−1))

m² ≤ 1

mE {B}²

+E(d_o(d_o−1))

m² ,

which yields that the second moment ofBis uniformly bounded by the square ofE(B): for any 0< α, E

B²

≤ E(do(do−1)) m−1

E(B)2

.

By (3.3), e^−αE(B)=E(B)−E(₁^B₊²_B), hence(1−e^−α)E(B)=E(₁^B₊²_B)≤E(B²)≤ ^E(do_m^(d₋^o₁⁻¹⁾⁾(E(B))².It follows that

E(B)≥ m(m−1) E(do(do−1))

1−e^−α .

On the other hand, by Jensen’s inequality,E(B)=e^αE₁₊^B_B ≤e^α_1+E^E(B)_(B),which implies that E(B)≤

e^α−1 .

Therefore,B/αis tight asα0. In particular, for some sub-sequenceα0,B(x)/αconverges in law to someY (x).

SinceB/αis bounded inL²uniformly inα >0 in a neighborhood of 0, we deduce from (3.2) that Y=^d 1

m N

i=1

Y_i, (3.5)

where N is distributed like do and, conditionally on N, (Yi) are i.i.d. and distributed as Y; moreover E(Y )= limα0E(^B_α) >0 (the limit along the same sub-sequence).

Dividing 1−e^−α

E(B)=E B²

1+B

≤E B²

byα², we get thatE(Y )=E(Y²). The same operation in (3.4) gives E(Y )²= m(m−1)

E(do(do−1))E Y²

. (3.6)

Putting these together we obtainE(Y )=^D_2m^o.

On the other hand, it is known (see e.g. [2], Theorem 16) that the law ofY satisfying (3.5) is determined up to a multiplicative constant, and thereforeY equals in distributionaW_ofor some constanta. The equalityEY=EY²then implies thatY equals in distribution W_o/E(W_o²). Since all possible limits in law are the same, we get thatβ(o)/α

converges in law toW_o/E(W_o²).

We return to the proof of the Einstein relation concerning velocities. Recall that alevel regeneration timeis a time for which the random walk hits a fresh level and never backtracks, see e.g. [3] for the definition and basic properties.

(Level regeneration times are related to, but different from, the regeneration times introduced in [17].) In particular, see [3], Section 4, and [19], Section 7, the differences of adjacent regeneration times form an i.i.d. sequence, with all moments bounded. Sinceγ_n(x)is smaller than thenth level regeneration time (started atx), it follows that the sequence γ_n(x)/n is uniformly integrable (under the measure GW×P_x,ω), and therefore, the convergence in the forthcoming (3.8) holds also in expectation:

nlim→∞

E[γn(o)]

n =E(β(o)) vα

. (3.7)

Sinceγ_n(x)=E_x,ω(τ_n1_(τ

n<T (^←x )))+O(1)and ^τ_nⁿ → _v¹_α, P_x,ω a.s. and inL¹(the latter follows at once from the integrability of regeneration times mentioned above, asτnis bounded above by thenth regeneration time), we get that forxfixed,

γ_n(x)

n →n→∞ 1 vα

P_x,ω

T (^←x )= ∞

=β(x) vα

, GWa.s. (3.8)

So all we need to prove in order to have the Einstein relation for velocities, is that

αlim→0 lim

n→∞

1 nE

γn(o)

= 1

m. (3.9)

To this end, define Bn(x):=1

λ

dx

i=1

βn(xi), Γn(x):=

dx

i=1

γn(xi), |x|< n.

Note that showing (3.9) is equivalent to proving that

αlim→0 lim

n→∞

1 nE

Γ_n(o)

=1. (3.10)

For|x|< n−1, Bn(x)=1

λ

dx

i=1

B_n(x_i)

1+B_n(x_i), Γn(x)=1 λ

dx

i=1

1+Γ_n(x_i)

1+B_n(x_i). (3.11)

Notice that we could defineΓ (x):=limn→∞Γn(x)

n , such that Γ (x):=1

λ

dx

i=1

Γ (x_i)

1+B(x_i). (3.12)

Asα→0 we can show thatΓ (x)→aY (x)for some constanta >0. The problem is that in the limit asn→ ∞we loose information on the value ofa(that should beE(do(do−1))/[m(m−1)]).

In order to determine this constant we have to make a step back and iterate Eq. (3.11) and, noticing thatΓn(x)=0 for all|x| =n−1, we get that

Γ_n(o)=

n−1

r=1

1 λ^r

|u|=r

1

1+B_n(u₁)· · · 1 1+B_n(u_r−1)

1

1+B_n(u_r):=

n−1

r=1

Φ_n(r), (3.13)

where {u₀, . . . , u_r} is the shortest path relating the root o to u [u0 =o,|u₁| =1, . . . ,|u_r| =r]. Note that B_n(u₁), . . . , B_n(u_r)are correlated.

Observe that Φ_n(r)≤e^αrW (o, r), consequently E(Φ_n(r))≤e^αr. Sinceα >0 it is hard to control the limit of Γ_n(o). The aim is to analyze the asymptotic behavior ofE(Φ_n(r))asn→ ∞andr≤n, which will be done in the following two subsections: in the next first subsection we will give a useful representation ofE(Φ_n(r))based on a spine random walk, whereas in the second subsection we make use of an argument from renewal theory and establish the limit ofE(Φ_n(r))whenr, n→ ∞in an appropriate way.

3.1. Spine random walk representation ofE(Φn(r))

LetΩ denote the space of rooted trees with no leaves. Denote byΩT the space of trees with a marked infinite ray Ray=(u^∗_n)n≥0, withu^∗₀=o[ΩT is topologically equal toΩT]. Unlike the setup used in Section2, where e.g.u^∗₁

was considered a parent ofo, we now redefine the notion of descendant inΩ_T. Namely, forx∈T,x =o, the parent ofx, denoted^←x, is the unique vertex on the geodesic connectingxandowith|^←x| = |x| −1. In this section, for any

|v|< n, we define the normalized progeny ofvat levelnasM_n(v):= |{w: |w| =n, wdescendant ofv}|/m^n−|v|, and M_n(v)=1 if|v| =n. We also writeM_n=M_n(o).

According to [14], on the spaceΩT we may construct a probabilityQsuch that the marginal ofQon the space of treesΩsatisfies

dQ dP

Fn

:=M_n, n≥1,

whereFnis theσ-field generated by the firstngenerations ofωandPdenotes the Galton–Watson law. Due top₀=0 and our tail assumptions, we have thatM_∞>0 a.s. and moreover,

Q

u^∗_n=u|Fn

= 1

M_nmⁿ ∀|u| =n.

UnderQ,d_u∗

n has the size-biased distribution associated with{p_k}, that isQ(d_u∗

n=k)=kp_k/m,u^∗_n+1is uniformly chosen among the children ofu^∗_nand, forv =u^∗_n+1with^←v =u^∗_n, the sub-treesT(v)rooted atv, are i.i.d. and have a Galton–Watson law.

For any 0≤j < n, we define a_j⁽ⁿ⁾:=¹_λ

v =u^∗_j+1,^←v=u^∗_jβ_n(v). Note that underQ, the family{a⁽ⁿ⁾_j }0≤j <n are in- dependent and eacha_j⁽ⁿ⁾is distributed as ¹_λd∗−1

k=1 β_n^(k)_−j−1, whered^∗has the size-biased distribution associated with {p_k},(β_l^(k), l≥1)are i.i.d. copies of(β_l(o), l≥1)and independent ofd^∗. We extenda_j⁽ⁿ⁾to allj∈Z∩(−∞, n−1]

by letting the family{a⁽ⁿ⁾_j , n > j}j∈Zbe independent (underQ) and such that for eachj,{a_j⁽ⁿ⁾, n > j}is distributed as {¹_λd∗−1

k=1 β_n^(k)_−j−1, n > j}. We naturally define a_j^(∞) as the limit of a_j⁽ⁿ⁾ as n→ ∞. In particular for j ≥0, a_j^(∞):=_λ¹

v =u^∗_j+1,^←v=u^∗_jβ(v), and each a_j^(∞) is distributed as ¹_λd∗−1

k=1 β^(k) with(β^(k))_k≥1 i.i.d. copies ofβ(o), independent ofd^∗.

The main result of this subsection is the following representation forE(Φn(r)):

Proposition 3.2. We may define a random walk(S_·, P )onZ,independent of the Galton–Watson treeωand of the family(a_j⁽ⁿ⁾)j <n,with step distributionP (Si−Si−1=1)=_λ+^λ_m2 andP (Si−Si−1= −1)=_λ+^m_m²2,∀i≥1,such that for any1≤r≤n,

E Φ_n(r)

=Q Zn(r)

Mn−r

, (3.14)

where

Z_n(r):=E_0,ω

1(τ_S(−r)<τ_S(n−r))

τ_S(−r)−1

i=0

f_n−r(S_i)

, 1≤r≤n,

withE0,ωthe expectation with respect to the random walkSstarting from0,and f_n(x):= m²+λ

m(1+λ+λa_x⁽ⁿ⁾)

, x < n. (3.15)

Before entering into the proof of Proposition3.2, we mention that the random walk(S_·, P )may find its root in the following lemma: