The most visited sites of biased random walks on trees

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Yueyun Hu, Zhan Shi

To cite this version:

Yueyun Hu, Zhan Shi. The most visited sites of biased random walks on trees. Electronic Journal of Probability, Institute of Mathematical Statistics (IMS), 2015, 20, pp.1-14. �10.1214/EJP.v20-4051�.

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El e c t ro nic J

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Electron. J. Probab.20(2015), no. 62, 1–14.

ISSN:1083-6489 DOI:10.1214/EJP.v20-4051

The most visited sites of biased random walks on trees

Yueyun Hu

Zhan Shi

Abstract

We consider the slow movement of randomly biased random walk(Xn)on a supercriti- cal Galton–Watson tree, and are interested in the sites on the tree that are most visited by the biased random walk. Our main result implies tightness of the distributions of the most visited sites under the annealed measure. This is in contrast with the one-dimensional case, and provides, to the best of our knowledge, the first non-trivial example of null recurrent random walk whose most visited sites are not transient, a question originally raised by Erd˝os and Révész [11] for simple symmetric random walk on the line.

Keywords:Biased random walk on the Galton–Watson tree; branching random walk; local time;

most visited site.

AMS MSC 2010:60J80 ; 60G50 ; 60K37.

Submitted to EJP on January 12, 2015, final version accepted on May 29, 2015.

SupersedesarXiv:1502.02831.

1 Introduction

We consider a (randomly) biased random walk(Xn)on a supercritical Galton–Watson treeT, rooted at∅. The random biases are represented byω:= (ω(x), x∈T\{∅}), a family of random vectors; for each vertexx∈T^,ω(x) := (ω(x, y), y ∈ T)is such that ω(x, y)≥0 for ally ∈T ^{and that}P

y∈Tω(x, y) = 1. For any vertex x∈T\{∅}, let^←x be its parent. For the sake of presentation, we modify the values ofω(∅, x)forxwith

←x=∅, and add a special vertex, denoted by^←∅, which is considered as the parent of∅^, such thatω(∅,^←∅) +P

x:^←x=∅ω(∅, x) = 1. The vertex^←∅is, however,notregarded as a vertex ofT; so, for example,P

x∈Tf(x)does not contain the termf(^←∅).

Assume that for each pair of verticesxandyinT∪ {^←∅},ω(x, y)>0if and only if y∼x, where byx∼ywe mean thatxis either a child, or the parent, ofy. Moreover, we defineω(^←∅,∅) := 1.

*Partly supported by ANR project MEMEMO2 (2010-BLAN-0125).

†LAGA, Institut Galilée, Université Paris XIII, 99 avenue J-B Clément, F-93430 Villetaneuse, France.

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

‡LPMA, Université Paris VI, 4 place Jussieu, F-75252 Paris Cedex 05, France.

E-mail:zhan.shi@upmc.fr

Givenω, the biased walk(Xn, n≥0) is a Markov chain taking values onT∪ {^←∅}, started atX₀=∅, whose transition probabilities are

Pω{Xn+1=y|Xn =x}=ω(x, y).

The probabilityP_ω is often referred to as the quenched probability. We also consider the annealed probabilityP(·) :=R

Pω(·)P(dω), wherePdenotes the probability with respect to the environment(ω,T).

There is an active literature on randomly biased walks on Galton-Watson trees; see, for example, a large list of references in [17]. In this paper, we restrict our attention to a regime ofslow movementof the walk in the recurrent case.

Clearly, the movement of the biased random walk(X_n)is determined by the law of the random environmentω. We assume that(ω(x, y), y ∼x)forx∈T, are i.i.d. random vectors. It is convenient to view(ω,T)as a marked tree (in the sense of Neveu [23]).

The influence of the random environment is quantified by means of the random potentialprocess(V(x), x∈T), defined byV(∅) := 0and

V(x) :=− X

y∈]]∅, x]]

log ω(^←y , y) ω(^←y ,^⇐y)

, x∈T\{∅}, (1.1)

where^⇐y is the parent of^←y, and ]]∅, x]] := [[∅, x]]\{∅}, with[[∅, x]]denoting the set of vertices (includingxand∅) on the unique shortest path connecting∅^tox. There exists an obvious bijection between the random environmentωand the random potentialV.

For anyx∈T^{, let}|x|denote its generation. Throughout the paper, we assume E X

x:|x|=1

e^−V(x)

= 1, E X

x:|x|=1

V(x) e^−V(x)

= 0. (1.2)

We also assume that the following integrability condition is fulfilled: there existsδ >0 such that

E X

x:|x|=1

e^{−(1+δ)V(x)}

+E X

x:|x|=1

e^δV(x)

+Eh X

x:|x|=1

1^1+δi

<∞. (1.3)

The random potential (V(x), x∈T)is a branching random walk as in Biggins [6];

as such, (1.2) corresponds to the “boundary case" (Biggins and Kyprianou [9]). It is known that, under some additional integrability assumptions that are weaker than (1.3), the branching random walk in the boundary case possesses some deep universality properties, see [25] for references.

Under(1.2)and(1.3), the biased walk(X_n)is null recurrent (Lyons and Pemantle [21], Menshikov and Petritis [22], Faraud [12]), such that upon the system’s survival,

|Xn| (logn)²

−→law X∞, (1.4)

1

(logn)³ max

0≤i≤n|X_i| → c₁ a.s., (1.5)

whereX∞is non-degenerate taking values in(0,∞), andc1denotes a positive constant:

bothX_∞andc₁are explicitly known, see [18] and [13], respectively.

For any vertexx∈T, let us define

Ln(x) :=

n

X

i=1

1_{Xi=x}, n≥1,

which is the (site) local time of the biased walk at x. Consider, for any n ≥ 1, the non-empty random set

An:=n

x∈T: Ln(x) = max

y∈TLn(y)o

. (1.6)

In words,Anis the set of the most visited sites (or: favourite sites) at timen. The study of favourite sites was initiated by Erd˝os and Révész [11] for the symmetric Bernoulli random walk on the line (see a list of ten open problems presented in Chapter 11 of the book of Révész [24]). In particular, for the symmetric Bernoulli random walk onZ^, Erd˝os and Révész [11] conjectured: (a) tightness for the family of most visited sites, and (b) the cardinality of the set of most visited sites being eventually bounded by2. Conjecture (b) was partially proved by Tóth [27], and is believed to be true by many. On the other hand, Conjecture (a) was disproved by Bass and Griffin [5]: as a matter of fact,inf{|x|, x∈An} → ∞almost surely for the one-dimensional Bernoulli walk. Later, we proved in [16] that it was also the case for Sinai’s one-dimensional random walk in random environment. The present paper is devoted to studying both questions for biased walks on trees; our answer is as follows.

Corollary 2.2.Assume(1.2)and(1.3). There exists a finite non-empty setUmin, defined in(2.5)and depending only on the environment, such that

n→∞lim P(An ⊂Umin|non-extinction) = 1. In particular, the family of most visited sites is tight underP^.

So, concerning the tightness question for most visited sites, biased walks on trees behave very differently from recurrent one-dimensional nearest-neighbour random walks (whether the environment is random or deterministic). To the best of our knowledge, this is the first non-trivial example of null recurrent Markov chain whose most visited sites are tight.

In the next section, we give a precise statement of the main result of this paper, Theorem 2.1.

2 Statement of results

Let us define a symmetrized version of the potential:

U(x) :=V(x)−log( 1 ω(x,^←x)

), x∈T. (2.1)

Note that

e^−U(x)= 1 ω(x,^←x)

e^−V(x)= e^−V(x)+ X

y∈T:^←y=x

e^−V(y), x∈T. (2.2)

It is known (Biggins [7], Lyons [20]) that under assumption (1.2), inf

x:|x|=nU(x)→ ∞, P^∗-a.s., (2.3) where here and in the sequel,

P^∗(·) := P(· |non-extinction), P^∗(·) := P(· |non-extinction).

Define thederivative martingale D_n := X

x:|x|=n

V(x)e^−V(x), n≥0. (2.4)

It is known (Biggins and Kyprianou [8], Aïdékon [1], Chen [10]) that (1.3) implies that D_n convergesP-a.s. to a limit, denoted byD_∞, and that

D_∞>0, P^∗-a.s.

Define the set of the minimizers ofU(·): Umin:=n

x∈T: U(x) = min

y∈TU(y)o

. (2.5)

Sinceinf_x:|x|=nU(x)→ ∞P^∗-a.s. (see (2.3)), the setUminis finite and non-empty.

The main result of the paper is as follows.

Theorem 2.1.Assume(1.2)and(1.3). For anyε >0,¹

sup

x∈T

Pω

n

Ln(x)

n logn

− σ²

4D_∞e^−U(x) > εo

→0, inP^∗-probability,

whereU(·)is the symmetrized potential in(2.1),D_∞theP^∗-almost sure positive limit of the derivative martingale(Dn)in(2.4), and

σ²:=E X

y:|y|=1

V(y)²e^−V(y)

∈(0,∞). (2.6)

Corollary 2.2.Assume(1.2)and(1.3). IfAnis the set of the most visited sites at timen as in(1.6), then

P^∗(An ⊂Umin)→1, whereUminis the set of the minimizers ofU(·)in(2.5).

Our results are not as strong as they might look like. For example, Theorem 2.1 does not claim thatP_ω{sup_x∈T|^Lnn^(x)

logn −_4D^σ2

∞e^−U(x)|> ε} →0inP^∗-probability. It essentially says, in view of Proposition 2.3 below, that for anyfixedx∈T^,P_ω{|^Lnn^(x)

logn

−_4D^σ2

∞e^−U(x)|>

ε} →0inP^∗-probability. Corollary 2.2 is much weaker than what Tóth [27] proved for the symmetric Bernoulli random walk onZ: for example, it does not claim thatP^∗-a.s., An⊂Uminfor all sufficiently largen; we even do not know whether this is true.

For local time at fixed site of biased random walks on Galton–Watson trees in other recurrent regimes, see the recent paper [15].

An important ingredient in the proof of Theorem 2.1 is the following estimate on the local time of vertices that are away from the root:

Proposition 2.3.Assume(1.2)and(1.3). Then

ε→0limlim sup

n→∞ P^∗n

max

x∈T:U(x)≥log(⁸

ε2)

L_n(x)≥ ε n logn

o

= 0.

In the light of the fact that inf_|x|=nU(x) → ∞ P^∗-a.s. (see (2.3)), Proposition 2.3 allows us, in the proof of Theorem 2.1, to estimate the probability for fixedx.

Proposition 2.3 is proved in Section 3; Theorem 2.1 and Corollary 2.2 in Section 4.

Throughout the paper, for any pair of verticesxandy, we writex < yory > xifyis a (strict) descendant ofx, andx≤yory≥xif eitheryis either a (strict) descendant of x, orxitself. For anyx∈T^{, we use}xi (for0≤i≤ |x|) to denote the ancestor ofxin the i-th generation; in particular,x₀=∅^andx_|x|=x.

1By convergence inP^∗-probability, we mean convergence in probability underP^∗.

3 Proof of Proposition 2.3

Before presenting the proof of Proposition 2.3, we outline the overall strategy. We exploit the relation between the local time atx, and the hitting timeT_xand the return timeT_∅⁺ (defined respectively in (3.4) and (3.5) below). Probabilities involving these random times are known to have standard one-dimensional formulas ((3.6) and (3.7)).

Proposition 2.3 is then proved using large deviation properties away from the mean if the potentialU(x)is large. SinceU(x)is indeedP^∗-a.s. large uniformly in the tree depth (as seen in (2.3)), we will be done.

We need some preliminaries. Let Λ(x) := X

y:^←y=x

e^{−[V(y)−V(x)]}, x∈T, (3.1)

In particular,Λ(∅) =P

x:|x|=1e^−V(x). By definition, 1 + Λ(x) = 1

ω(x,^←x).

LetSi−S_i−1,i≥1, be i.i.d. random variables whose law is characterized by Eh

h(S1)i

=Eh X

x∈T:|x|=1

e^−V(x)h(V(x))i

, (3.2)

for any Borel functionh: R→R+.

The following fact, quoted from [18], is a variant of the so-called “many-to-one formula" for the branching random walk.

Fact 3.1.Assume(1.2)and(1.3). LetΛ(x)be as in(3.1). For anyn≥1and any Borel functiong:Rⁿ⁺¹→R+, we have

Eh X

x∈T:|x|=n

g

V(x1),· · · , V(xn),Λ(x)i

=Eh e^SnG

S1,· · · , Sn

i ,

whereSi−Si−1,i≥1, are i.i.d. whose common distribution is given in(3.2), and

G(a1,· · · , an) :=E[g(a1,· · · , an, X

x∈T:|x|=1

e^−V(x))].

Define a reflecting barrier at (notation:]]∅, x[[ := ]]∅, x]]\{x}) Ln^(γ) := n

x: X

z∈]]∅, x]]

e^V(z)−V(x)> n (logn)^γ,

X

z∈]]∅, y]]

e^V(z)−V(y)≤ n

(logn)^γ, ∀y∈]]∅, x[[o

, (3.3)

whereγ∈Ris a fixed parameter. We writex <Ln^(γ)ifP

z∈]]∅, y]]e^V(z)−V(y)≤_(logⁿ_n)γ for ally∈]]∅, x[[.

We recall two results from [18]. The first justifies the presence of the barrierLn^(γ)

for the biased walk(Xn), and the second describes the local time at the root.

Fact 3.2([18]).Assume(1.2)and(1.3). Ifγ <2, then

n→∞lim P[ⁿ

i=1

{Xi∈Ln^(γ)}

= 0.

Fact 3.3([18]).Assume(1.2)and(1.3). For anyε >0,

P_ωn

L_n(∅)

n logn

− σ² 4D∞

e^−U(∅) > εo

→0, inP^∗-probability.

We now proceed to the proof of Proposition 2.3. Define

T_x := inf{i≥0 : X_i=x}, x∈T, (3.4)

T_∅⁺ := inf{i≥1 : Xi=∅}. (3.5)

In words,Tx is the first hitting time atxby the biased walk, whereasT_∅⁺ is the first returntime to the root∅^.

Let x ∈ T\{∅}. The probability Pω(Tx < T_∅⁺) only involves a one-dimensional random walk in random environment (namely, the restriction at[[∅, x[[ of the biased walk (X_i)), so a standard result for one-dimensional random walks in random environment (Golosov [14]) tells us that

Pω(Tx< T_∅⁺) = ω(∅, x₁) e^V(x1) P

z∈]]∅, x]]e^V(z) = ω(∅,^←∅) P

z∈]]∅, x]]e^V(z), (3.6) Px,ω{T_∅< T_x⁺} = e^U(x)

P

z∈]]∅, x]]e^V(z), (3.7)

wherex1is the ancestor ofxin the first generation.

Proof of Proposition 2.3.By Fact 3.2, for allγ₁<2, we haveP^∗(∪^m_i=1{Xi∈L^m(γ1)})→0, m→ ∞. So it suffices to check that for someγ₁<2,

b→0limlim sup

m→∞ P^∗n

max

x<Lm^{(γ1 )}:U(x)≥log(⁸

b2)

Lm(x)≥ bm logm

o

= 0.

SinceP^∗(U(∅)≥log(_b⁸2))→0forb→0, it suffices to prove that for someγ1<2, lim

b→0lim sup

m→∞ P^∗n

max

x∈T\{∅}:x<Lm^{(γ1 )}, U(x)≥log(⁸

b2)

L_m(x)≥ bm logm

o

= 0. (3.8)

LetT_∅⁽⁰⁾ := 0and inductivelyT_∅^(j) := inf{i > T_∅^(j−1): X_i =∅}, forj ≥1. In words, T_∅^(j)is thej-th return time to ∅. We have, forn ≥2, c >0, ε∈(0,1), 1 < γ <2and m(n) =bc nlognc,

P^∗n

max

x∈T\{∅}:x<L_m(n)^(γ) , U(x)≥log(⁸_ε)

L_m(n)(x)≥εno

≤ P^∗{T_∅⁽ⁿ⁾≤m(n)}+P^∗n

max

x∈T\{∅}:x<L_m(n)^(γ) , U(x)≥log(⁸_ε)

L_T(n)

∅

(x)≥εno .

By Fact 3.3, ^T

(n)

∅

nlogn → ^4D_σ2^∞e^U(∅)inP^∗-probability, so the portmanteau theorem implies thatlim sup_n→∞P^∗{T_∅⁽ⁿ⁾≤m(n)} ≤P^∗{^4D_σ2^∞e^U(∅)≤c}. Assume, for the time being, that we are able to prove that for someγ <2, anyc >0and any0< ε <1,

P^∗n

max

x∈T\{∅}:x<L_m(n)^(γ) , U(x)≥log(⁸_ε)

L_T(n)

∅

(x)≥εno

→0, n→ ∞. (3.9)

Then we will have lim sup

n→∞ P^∗n

max

x∈T\{∅}:x<L_m(n)^(γ) , U(x)≥log(⁸_ε)

L_m(n)(x)≥εno

≤P^∗n4D_∞

σ² e^U(∅)≤co .

Sincen≤²_clog^m(n+1)_m(n) (for all sufficiently largen), this will yield

lim sup

n→∞ P^∗n

max

x∈T\{∅}:x<L_m(n)^(γ) , U(x)≥log(⁸_ε)

L_m(n)(x)≥ 2ε c

m(n+ 1) logm(n)

o≤P^∗n4D∞

σ² e^U(∅)≤co .

Letm∈[m(n), m(n+ 1)]∩Z^{. Then}L_m(n)(x)≤Lm(x)(for allx∈T); on the other hand, ifx <Lm^(γ1), thenx <L_m(n)^{(γ) for all}γ₁∈(γ,2)and all sufficiently largen. Consequently, we will have, for allc >0andε∈(0, 1),

lim sup

m→∞ P^∗n

max

x∈T\{∅}:x<Lm^{(γ1 )}, U(x)≥log(⁸_ε)

Lm(x)≥2ε c

m logm

o≤P^∗n4D_∞

σ² e^U(∅)≤co .

Takingc:= 2ε^1/2will then yield (3.8) (writingb:=ε^1/2there) and thus Proposition 2.3.

The rest of the section is devoted to the proof of (3.9). By (1.5), _(log¹_n)3max_0≤i≤n|Xi| convergesP^∗-a.s. to a positive constant, and since ^T

(n)

∅

nlogn converges inP^∗-probability to a positive limit, we deduce that _(log¹_n)3 max_0≤i≤T(n)

∅

|Xi|converges inP^∗-probability to a positive limit. So the proof of (3.9) is reduced to showing the following estimate: for some1< γ <2, anyc >0and any0< ε <1,

P^∗n

max

x<L_m(n)^(γ) :U(x)≥log(⁸_ε),1≤|x|≤(logn)⁴

L_T(n)

∅

(x)≥εno

→0, n→ ∞.

Fork≥1, we have Pω

n

max

x<Lm(n)^(γ) :U(x)≥log(⁸_ε),1≤|x|≤(logn)⁴

L_T(n)

∅

(x)≥ko

≤ X

x<Lm(n)^(γ) :U(x)≥log(⁸_ε),1≤|x|≤(logn)⁴

P_ω{L_T(n)

∅

(x)≥k}.

The law ofL

T_∅⁽ⁿ⁾(x)underP_ωis the law ofPn

i=1ξ_i, where(ξ_i, i≥1)is an i.i.d. sequence withP_ω(ξ₁= 0) = 1−aandP_ω(ξ₁≥k) =a p^k−1,∀k≥1, where

1−p := P_x,ω{T_∅ < T_x⁺}= e^U(x) P

z∈]]∅, x]]e^V(z), (3.10) a := P_ω{T_x< T_∅⁺}= ω(∅,^←∅)

P

z∈]]∅, x]]e^V(z). (3.11) [We have used (3.6) and (3.7).]

The tail estimate of L_T(n)

∅

(x)underPω is summarized in the following elementary lemma, whose proof is in the Appendix.

Lemma 3.4.Let0< a <1and0< p <1. Let(ξi, i≥1)be an i.i.d. sequence of random variables withP(ξ₁= 0) = 1−aandP(ξ₁≥k) =a p^k−1,∀k≥1.

Let0< ε <1. If1−p > ⁸_εa, then

PnXⁿ

i=1

ξi≥ dεneo

≤6nae^{−(1−p)εn8} .

We continue with the proof of (3.9). IfU(x)≥log(⁸_ε), then1−p > ⁸_εa, so we are entitled to apply Lemma 3.4 to arrive at:

P_ωn

max

x<L_m(n)^(γ) :U(x)≥log(⁸_ε),1≤|x|≤(logn)⁴

L_T(n)

∅

(x)≥ dεneo

≤ 6n X

x<Lm(n)^(γ) : 1≤|x|≤(logn)⁴

ω(∅,^←∅) P

z∈]]∅, x]]e^V(z)exp

−εn 8

e^U(x) P

z∈]]∅, x]]e^V(z) .

We haveω(∅,^←∅)≤1. It remains to check the following convergence inP^∗-probability (forn→ ∞):

X

x<L_m(n)^(γ) : 1≤|x|≤(logn)⁴

n P

z∈]]∅, x]]e^V(z)exp

−ε 8

ne^U(x) P

z∈]]∅, x]]e^V(z)

→0. (3.12)

Recall the definition ofL_m(n)^{(γ) :} x < L_m(n)^{(γ) implies P eV(x)}

z∈]]∅, x]]e^V(z) ≥ ^(log_m(n)^m(n))γ, which is

≥ ^(log_cn^n)γ−1 for all sufficiently largen(sayn≥n0). Also, we recall thate^U(x)= _1+Λ(x)^eV(x) , withΛ(x) :=P

y:^←y=xe^{−[V(y)−V(x)]}as in (3.1).

For the sumP

x<L_m(n)^(γ) on the left-hand side of (3.12), we distinguish two possible situations depending on the value of Λ(x). Let 0 < % < 1. Applying the elementary inequalityλe^−λ≤c2e^−λ/2(forλ≥0) toλ:=^ε₈ P ^neU(x)

z∈]]∅, x]]e^V(z), we see that forn≥n0, X

x<L_m(n)^(γ)

1{1+Λ(x)≤(P ^neV(x) z∈]]∅, x]] eV(z))^%}

n P

z∈]]∅, x]]e^V(z)exp

−ε 8

ne^U(x) P

z∈]]∅, x]]e^V(z)

≤ c₂ X

x<Lm(n)^(γ)

1{1+Λ(x)≤(P ^neV(x) z∈]]∅, x]] eV(z))^%}

8

εe^−U(x)exp

− ε

16(1 + Λ(x))

ne^V(x) P

z∈]]∅, x]]e^V(z)

≤ 8c2

ε

X

x<L_m(n)^(γ)

e^−U(x)exp

− ε

16( ne^V(x) P

z∈]]∅, x]]e^V(z))^1−%

.

Since P ^neV(x)

z∈]]∅, x]]e^V(z) ≥¹_c(logn)^γ−1(forx <L_m(n)^{(γ) and}n≥n0), this yields, forn≥n0, X

x<L_m(n)^(γ)

1{1+Λ(x)≤(P ^neV(x) z∈]]∅, x]] eV(z))^%}

n P

z∈]]∅, x]]e^V(z)exp

−ε 8

ne^U(x) P

z∈]]∅, x]]e^V(z)

≤ 8c2

ε exp

− ε

16c^1−%(logn)^{(γ−1)(1−%)} X

x<L_m(n)^(γ)

e^−U(x),

which converges to0inP^∗-probability (recalling that for anyγ∈R^,log¹n

P

x∈T:x<Ln^(γ)e^−U(x) converges inP^∗-probability to a finite limit; see [18]). So it remains to prove that there exists%∈(0, 1)such that (removing the big exponential term which is bounded by1)

X

x<L_m(n)^(γ) : 1≤|x|≤(logn)⁴

1{1+Λ(x)>(_P ^neV(x)

z∈]]∅, x]] eV(z))^%}

n P

z∈]]∅, x]]e^V(z) →0,

in P^∗-probability (for n → ∞). Sincelim_r→∞inf_|x|=rV(x) → ∞P^∗-a.s. (see (2.3)), it suffices to prove the existence of%∈(0,1) andγ ∈ (1,2)such that for all α >0and

n→ ∞, X

x<L_m(n)^(γ) : 1≤|x|≤(logn)⁴

1{1+Λ(x)>(P ^neV(x) z∈]]∅, x]] eV(z))^%}

n P

z∈]]∅, x]]e^V(z)1_{{V(x)≥−α}}→0, (3.13) inP^∗-probability, whereV(x) := min_{z∈]]∅, x]]}V(z).

To prove this, we first recall thatx <L_m(n)^(γ) implies that for ally ∈]]∅, x]], we have

P

z∈]]∅, y]]e^V(z)

e^V(y) ≤ _(log^cn_n)γ−1 (forn≥n0) which is bounded bynfor all sufficiently largen (sayn≥n₁); a fortioriV(y)−V(y)≤logn(withV(y) := max_{z∈]]∅, y]]}V(y)). By Fact 3.1, we obtain, forn≥n₀∨n₁,

Eh X

x<L_m(n)^(γ) ,1≤|x|≤(logn)⁴

1{1+Λ(x)>(_P ^neV(x)

z∈]]∅, x]] eV(z))^%}

n P

z∈]]∅, x]]e^V(z)1_{{V(x)≥−α}}i

≤

b(logn)⁴c

X

k=1

Eh X

x:|x|=k

1_{{V(y)−V(y)≤logn,∀y∈]]∅, x]]}}×

1{1+Λ(x)>(_P ^neV(x)

z∈]]∅, x]] eV(z))^%}

n P

z∈]]∅, x]]e^V(z)1_{{V(x)≥−α}}i

=

b(logn)⁴c

X

k=1

Eh

e^Sk1_{S#

k≤logn}F

( ne^Sk Pk

i=1e^Si)^% n Pk

i=1e^Si 1_{S

k≥−α}

i ,

whereF(λ) :=P(1+P

x:|x|=1e^−V(x)> λ)forλ >0,Sk := max1≤i≤kSi,S_k:= min1≤i≤kSi, andS^#_k := max1≤i≤k(Si−Si)for anyk≥1.

An application of the Hölder inequality, using assumption (1.3), yields the existence ofδ₁>0such that

Eh X

x:|x|=1

e^−V(x)1+δ1i

<∞. (3.14)

As such, c3 := E[(1 +P

x:|x|=1e^−V(x))^1+δ1] < ∞, so F(λ) ≤ c3λ^−1−δ1 for all λ > 0. Consequently,

Eh X

x<L_m(n)^(γ) ,1≤|x|≤(logn)⁴

1{1+Λ(x)>(P ^neV(x) z∈]]∅, x]] eV(z))^%}

n P

z∈]]∅, x]]e^V(z)1_{{V(x)≥−α}}i

≤c3

b(logn)⁴c

X

k=1

EhPk i=1e^Si ne^Sk

^%(1+δ1)−1

1_{S#

k≤logn}1_{S

k≥−α}

i

. (3.15)

Lemma 3.5.Letδbe the constant in assumption(1.3). For allα >0andδ2∈(0, δ∧₁₆¹),

n→∞lim

b(logn)⁴c

X

k=1

EhPk i=1e^Si ne^Sk

^δ2

1_{S#

k≤logn}1_{S

k≥−α}

i

= 0.

Since it is possible to choose0< % <1such that%(1 +δ₁)−1lies in(0, δ∧₁₆¹), we can apply Lemma 3.5 to see that (3.15) implies (3.13), and thus yields Proposition 2.3.

It remains to prove Lemma 3.5.

Proof of Lemma 3.5.SincePk

i=1e^Si ≤ke^Sk, it suffices to check that (logn)^4δ2

n^δ2

b(logn)⁴c

X

k=1

Eh

e^δ2(Sk−Sk)1_{S#

k≤logn}1_{S

k≥−α}

i→0.

Recall the law ofS₁from (3.2). By assumption (1.3) and Hölder’s inequality, we have E(e^aS1)<∞, ∀a∈(−δ,1 +δ),

whereδ >0is the constant in (1.3). In particular,E(e^a|S1|)<∞for all0≤a < δ. Since 0 < δ2 < δ, we haveE(e^δ2(Sk−Sk)) ≤e^c4k for some constantc4 > 0 and allk ≥1. So

(logn)^4δ2 n^δ2

Pb(logn)^1/2c

k=1 E[e^δ2(Sk−Sk)]→0. It remains to prove that (logn)^4δ2

n^δ2

b(logn)⁴c

X

k=b(logn)^1/2c

Eh

e^δ2(Sk−Sk)1_{S#

k≤logn}1_{S

k≥−α}

i→0.

We make a change of indicesk=b(logn)^1/2c+`. LetSe`:=S_{`+b(logn)</sub>1/2c−S<sub>b(logn)</sub>1/2c,`≥ 0. Then(Se`, `≥0)is a random walk having the law of(S`, `≥0), and is independent of (Si,1≤i≤ b(logn)^1/2c). For`≥0,S<sub>`+b(logn)}1/2c−S_{`+b(logn)</sub>1/2c = max(x,max<sub>0≤j≤`}Sej)−

Se` ≥ max<sub>0≤j≤`</sub>Sej −Se`, where x := S<sub>b(logn)</sub>1/2c −S<sub>b(logn)</sub>1/2c. So for k ≥ b(logn)<sup>1/2</sup>c and` := k− b(logn)^1/2c, on the event that{S_k ≥ −α}, eithermax_{0≤j≤`</sub>Sej ≤ x, then S<sub>k</sub>−S<sub>k</sub> =x−Se<sub>`}=S_b(logn)1/2c−S_k≤S_b(logn)1/2c+α, ormax<sub>0≤j≤`</sub>Se<sub>j</sub>> x, thenS<sub>k</sub>−S<sub>k</sub>= max0≤j≤`Sej−Se`. It follows that

Eh

e^δ2(Sk−Sk)1_{S#

k≤logn}1_{S

b(logn)1/2c≥−α}

i

≤ E(e^{δ2(α+Sb(logn)1/2c)}) +P(S_b(logn)1/2c≥ −α)×Eh

e^δ2(S`−S`)1_{S#

` ≤logn}

i .

SinceE(e^{δ2Sb(logn)1/2c})≤e^c4(logn)1/2, we have^(log_nⁿ⁾δ2^4δ2

Pb(logn)⁴c

k=b(logn)^1/2cE(e^{δ2(α+Sb(logn)1/2c)})→ 0. On the other hand,P(S_b(logn)1/2c ≥ −α)≤c5(logn)^−1/4for some constantc5>0and alln≥2(see Kozlov [19]); it suffices to prove that

(logn)^4δ2−(1/4) n^δ2

∞

X

`=0

Eh

e^δ2(S`−S`)1_{S#

`≤logn}

i→0.

This will be a straightforward consequence of the following estimate (applied toλ:= logn andb:=δ2; it is here we use the conditionδ2< ₁₆¹): for any0< b < δ,

lim sup

λ→∞

E^τX^λ−1

`=0

e^{−b[λ−(S`−S`)]}

<∞, (3.16)

whereτλ:= inf{i≥1 : Si−Si> λ}.

To prove (3.16), we define the (strictly) ascending ladder times(Hi, i≥0):H0:= 0 and for anyi≥1,

H_i:= inf{` > H<sub>i−1</sub>:S<sub>`</sub>> max

0≤j≤Hi−1

S_j}.

Therefore,

E^τX^λ−1

`=0

e^b(S`−S`)

=

∞

X

i=1

E ^HXⁱ⁻¹

`=Hi−1

e^{b(SHi−1−S`)}1_{S#

` ≤λ}

.

We apply the strong Markov property, first at timeH_i−1to see that

E ^HXⁱ⁻¹

`=Hi−1

e^{b(SHi−1−S`)}1_{S#

`≤λ}

≤P S_H^#

i−1 ≤λ

E^HX¹⁻¹

`=0

e^−bS`1_{S#

` ≤λ}

,

and then successively at timesH₁,H₂, · · ·,H_i−1to see thatP(S_H^#

i−1 ≤λ)≤P(S_H^#

1 ≤

λ)ⁱ⁻¹. As such,

E^τX^λ−1

`=0

e^b(S`−S`)

≤

∞

X

i=1

P S^#_H

1 ≤λi−1

E^HX¹⁻¹

`=0

e^−bS`1_{S#

`≤λ}

.

We defineσ−λ:= inf{n≥0 :Sn<−λ}. Then1−P(S_H^#

1 ≤λ) =P(σ−λ< H1)≥ _1+λ^c6 for some constantc₆ >0 and all sufficiently largeλ, sayλ≥λ₀ (for the last elementary inequality, see for example, Lemma A.1 in [17]). Thus we get that

E^τX^λ−1

`=0

e^b(S`−S`)

≤ 1 +λ

c₆ E^HX¹⁻¹

`=0

e^−bS`1_{S#

`≤λ}

.

Finally, for all smallb >0, there exists some positive constantc₇=c₇(b)>0such that

E^HX¹⁻¹

`=0

e^−bS`1_{S#

`≤λ}

=E^HX¹⁻¹

`=0

e<sup>−bS`</sup>1<sub>{σ−λ>`}</sub>

≤ c7

λ e^bλ,

by applying [2] (Lemma 6, formula (4.17)) to (−S_i, i ≥ 1). This yields (3.16), and

completes the proof of Lemma 3.5 and Proposition 2.3. 2

4 Proof of Theorem 2.1 and Corollary 2.2

Proof of Theorem 2.1. Recall thatlim_k→∞inf_x:|x|=kU(x) → ∞ P^∗-a.s. (see (2.3)). In view of Proposition 2.3, we only need to prove that for any fixedx∈T^andε >0, when n→ ∞,

Pω

n

Ln(x)

n logn

− σ²

4D_∞e^−U(x) > εo

→0, inP^∗-probability.

According to Fact 3.3, this is equivalent to convergence inP^∗-probabilityP_ω{ |_L^Ln(x)

n(∅)− e^{−[U(x)−U(∅)]}|> ε} →0(forn→ ∞), and thus to the following statement: for anyx∈T andm→ ∞,

P_ωn

L_T(m)

∅

(x)

m −e^{−[U(x)−U(∅)]}

> εo

→0, inP^∗-probability,

whereT_∅^(m)is as before, them-th return time of the biased walk(Xi)to the root∅^{. This,} however, holds trivially asL_T(m)

∅

(x)−L_T(m−1)

∅

(x),m≥1, are i.i.d. random variables under PωwithEω[L_T(1)

∅

(x)] = _1−p^a = e^{−[U(x)−U(∅)]}(see the notation at (3.10)–(3.11) as well as the discussion preceding the equations). Theorem 2.1 is proved. 2 Proof of Corollary 2.2.Letε >0and0< a < ¹₂. Let

En(ε, a) :=n ω: sup

x∈T

Pω

Ln(x)

n logn

− σ²

4D_∞e^−U(x) > ε

< ao .

By Theorem 2.1,P^∗(En(ε, a))→1,n→ ∞.

Letx_n ∈An, and letx_min∈Umin. For allω∈E_n(ε, a), we have

Pω

Ln(y)

n logn

− σ²

4D_∞e^−U(y) ≤ε

≥1−a ,

fory=x_nand fory=x_min; hence, for allω∈E_n(ε, a),

Pω

Ln(xn)

n logn

≤ σ²

4D_∞e^−U(xn)+ε, Ln(xmin)

n logn

≥ σ²

4D_∞e^−U(xmin)−ε

≥1−2a .

By definition,Ln(xn) = sup_x∈TLn(x)≥Ln(xmin). Therefore, for allω,

Pω

σ²

4D_∞e^−U(xn)≥ σ²

4D_∞e^−U(xmin)−2ε

≥(1−2a)1_{En(ε, a)}(ω).

Taking expectation with respect toP^∗on both sides gives that

P^∗ σ²

4D_∞e^−U(xn)≥sup

x∈T

σ²

4D_∞e^−U(x)−2ε

≥(1−2a)P^∗(En(ε, a)),

which converges to1−2awhenn→ ∞. Sincea >0can be as small as possible, this yields

σ²

4D_∞ e^−U(xn)→sup_x∈T4D^σ2

∞e^−U(x)in probability underP^{∗, i.e.,}U(x_n)→inf_x∈TU(x)in probability underP^∗.

SinceUmin, the set of the minimizers ofU(·), isP^∗-a.s. finite, we haveinf_x∈UminU(x)<

inf_x∈T\UminU(x)P^∗-a.s., which yieldsP^∗(xn ∈Umin)→1,n→ ∞. 2

A Appendix: Proof of Lemma 3.4

Lets∈[1, ¹_p). ThenE(s^ξ1) = 1−a+^a(1−p)s_1−ps . So

PnXⁿ

i=1

ξi≥ko

≤ 1 s^k Eh

s^Pni=1ξi1_{^Pn

i=1ξi>0}

i

= [E(s^ξ1)]ⁿ−[P{ξ1= 0}]ⁿ

s^k .

Observe that

[E(s^ξ1)]ⁿ−[P{ξ1= 0}]ⁿ =

1−a+a(1−p)s 1−ps

ⁿ

−(1−a)ⁿ

≤ na(1−p)s 1−ps

1−a+a(1−p)s 1−ps

n−1

,

where, in the last line, we usedxⁿ−yⁿ≤n(x−y)xⁿ⁻¹(for0≤y≤x). Hence

PnXⁿ

i=1

ξi≥ko

≤s^−kna(1−p)s 1−ps

1−a+a(1−p)s 1−ps

n−1

. (A.1)

First case: ¹₃≤p <1. We takes:= ^1+p_2p ∈[1, ¹_p), so that ^(1−p)s_1−ps = ^1+p_p ; hence by (A.1),

PnXⁿ

i=1

ξi≥ko

≤ 1 +p 2p

−k

na(1 +p) p

1 +a

p n−1

= na1 +p p

1 +1−p 2p

−k 1 + a

p n−1

≤ 2na p

1 + 1−p 2p

^−k 1 +a

p ⁿ

.

Since(1 +u)⁻¹≤e^−u/2(for0≤u≤1) and1 +v≤e^v(forv≥0), applied tou:= ^1−p_2p ≤1 andv:= ^a_p, we obtain, in case ¹₃ ≤p <1,

PnXⁿ

i=1

ξi≥ko

≤6naexp

−(1−p)k 4p +na

p

.