The discrete-time parabolic Anderson model with heavy-tailed potential

www.imstat.org/aihp 2012, Vol. 48, No. 4, 1049–1080

DOI:10.1214/11-AIHP465

© Association des Publications de l’Institut Henri Poincaré, 2012

The discrete-time parabolic Anderson model with heavy-tailed potential

Francesco Caravenna

Philippe Carmona

and Nicolas Pétrélis

aDipartimento di Matematica e Applicazioni, Università degli Studi di Milano-Bicocca, via Cozzi 53, 20125 Milano, Italy.

E-mail:[email protected]

bLaboratoire de Mathématiques Jean Leray UMR 6629, Université de Nantes, 2 Rue de la Houssinière, BP 92208, F-44322 Nantes Cedex 03, France. E-mail:[email protected]

cLaboratoire de Mathématiques Jean Leray UMR 6629, Université de Nantes, 2 Rue de la Houssinière, BP 92208, F-44322 Nantes Cedex 03, France. E-mail:[email protected]

Received 21 December 2010; revised 27 October 2011; accepted 15 November 2011

Abstract. We consider a discrete-time version of the parabolic Anderson model. This may be described as a model for a directed (1+d)-dimensional polymer interacting with a random potential, which is constant in the deterministic direction and i.i.d. in the dorthogonal directions. The potential at each site is a positive random variable with a polynomial tail at infinity. We show that, as the size of the system diverges, the polymer extremity is localized almost surely at one single point which grows ballistically. We give an explicit characterization of the localization point and of the typical paths of the model.

Résumé. Nous considérons une version discrète du modèle parabolique d’Anderson. Ceci nous permet, par exemple, d’étudier un polymère dirigé en dimension 1+dqui interagit avec un potentiel constant dans la direction déterministe et i.i.d. dans l’hyperplan orthogonal. Le potentiel en chaque site est une variable aléatoire positive dont la queue décroît polynomialement. Nous prouvons que, lorsque la taille du système tend vers l’infini, l’extrémité du polymère se localise presque surement en un site unique, que nous caractérisons et qui s’éloigne balistiquement de l’origine. Nous donnons également une caractérisation du comportement typique des trajectoires de ce modèle.

MSC:60K37; 82B44; 82B41

Keywords:Parabolic Anderson model; Directed polymer; Heavy tailed potential; Random environment; Localization

1. Introduction and results

The model we consider is built on two main ingredients, a random walkSand a random potentialξ. We start describing these ingredients. A word about notation: throughout the paper, we denote by| · |the¹norm onR^d, that is |x| =

|x₁| + · · · + |x_d|forx=(x₁, . . . , x_d), and we setBN:= {x∈Z^d: |x| ≤N}. 1.1. The random walk

LetS= {S_k}k≥0denote the coordinate process on the spaceΩ_S:=(Z^d)^{N0:={0,1,2,...}}, that we equip as usual with the product topology andσ-field. We denote by P the law onΩ_Sunder whichSis a (lazy) nearest-neighbor random walk

1Supported by the University of Padova under Grant CPDA082105/08.

2Supported by the Grant ANR-2010-BLAN-0108.

started at zero, that is P(S0=0)=1 and under P the variables{S_k+1−S_k}k≥0are i.i.d. with P(S1=y)=0 if|y|>1.

We also assume the following irreducibility conditions:

P(S1=0)=:κ >0 and P(S1=y) >0 ∀y∈Z^dwith|y| =1. (1.1) The usual assumption E(S1)=0 is not necessary. Forx∈Z^d, we denote by Pxthe law of the random walk started at x, that is Px(S∈ ·):=P(S+x∈ ·).

We could actually deal with random walks with finite range, i.e., for which there existsR >0 such that P(S1= y)=0 if|y|> R, but we stick for simplicity to the caseR=1.

1.2. The random potential

We letξ = {ξ(x)}x∈Z^d denote a family of i.i.d. random variables taking values inR⁺, defined on some probability space(Ωξ,F,P), which should not be confused withΩS. We assume that the variablesξ(x)are Pareto distributed, that is

P

ξ(0)∈dx

= α

x^1+α1[1,∞)(x)dx (1.2)

for someα∈(0,∞). Although the precise assumption (1.2) on the law ofξ could be relaxed to a certain extent, we prefer to keep it for the sake of simplicity.

In the sequel we could work with the product spaceΩ_S×Ω_ξ, equipped with the product probability P⊗P, under whichξ andSare independent, but it is actually not necessary, becauseξ andSwill act on a separate level, as it will be clear in a moment.

1.3. The model

GivenN∈N:= {1,2,3, . . .}and aP-typical realization of the variablesξ = {ξ(y)}y∈Z^d, our model is the probability PN,ξ onΩ_S defined by

dPN,ξ

dP (S):= 1

U_N,ξe^HN,ξ(S), whereHN,ξ(S):=

N i=1

ξ(Si) (1.3)

and the normalizing constantU_N,ξ (partition function) is of course UN,ξ :=E

e^HN,ξ(S)

=E

exp _N

i=1

ξ(Si) . (1.4)

We stress that we are dealing with aquenched disordered model: we are interested in the properties of the lawPN,ξ

forP-typical butfixedrealizations of the potentialξ.

Let us also introduce theconstrained partition functionu_N,ξ(x), defined forx∈Z^dby u_N,ξ(x):=E

exp

_N

i=1

ξ(S_i) 1_{SN=x}

(1.5) so thatU_N,ξ=

x∈Z^du_N,ξ(x). Note that the law ofS_NunderPN,ξ is given by p_N,ξ(x):=PN,ξ(S_N=x)=u_N,ξ(x)

U_N,ξ = u_N,ξ(x)

y∈Z^du_N,ξ(y). (1.6)

The law PN,ξ admits the following interpretation: the trajectories {(i, S_i)}0≤i≤N model the configurations of a(1+d)-dimensional directed polymer of length N which interacts with the random potential (or environment) {ξ(x)}x∈Z^d. The random variableξ(x)should be viewed as a reward sitting at sitex∈Z^d, so that the energy of each

polymer configuration is given by the sum of the rewards visited by the polymer. On an intuitive ground, the polymer configurations should target the sites where the potential takes very large values. The corresponding energetic gain entails of course an entropic loss, which however should not be too relevant, in view of the heavy tail assumption (1.2).

As we are going to see, this is indeed what happens, in a very strong form.

Besides the directed polymer interpretation, PN,ξ is a law onΩS =(Z^d)^N0 which may be viewed as a natural penalization of the random walk law P. In particular, when looking at the process{S_k}k≥0under the lawPN,ξ, we often considerkas a time parameter.

Remark 1.1. An alternative interpretation of our model is to describe the spatial distribution of a population evolving in time.At time zero,the population consists of one individual located at the sitex=0∈Z^d.In each time step,every individual in the population performs one step of the random walkS,independently of all other individuals,jumping from its current sitex to a sitey (possiblyy=x)and then splitting into a number of individuals(always at sitey) distributed like aPo(e^ξ(y)),wherePo(λ)denotes the Poisson distribution of parameterλ >0.The expected number of individuals at sitex∈Z^d at timeN∈Nis then given byuN,ξ(x),as one checks easily.

Remark 1.2. Our model is somewhat close in spirit to the much studieddirected polymer in random environment [2,3,9],in which the rewardsξ(i, x)depend also oni∈N(and are usually chosen to be jointly i.i.d.).In our model, the rewards are constant in the “deterministic direction”(1,0),a feature which makes the environment much more attractive from a localization viewpoint.Notice in fact that a sitexwith a large rewardξ(x)yields a favorable straight corridor{0, . . . , N} × {x}for the polymer{(i, S_i)}0≤i≤N.

We also point out that the so-calledstretched polymer in random environmentwith a fixed length,considered e.g.in [7],is a model which provides an interpolation between the directed polymer in random environment and our model.

1.4. The main results

The closest relative of our model is obtained considering the continuous-time analoguˆ_t,ξ(x)of (1.5), defined for t∈ [0,∞)andx∈Z^dby

ˆ

u_t,ξ(x):=E

exp t

0

ξ(Sˆ_u)du

1_{{ ˆS}

t=x}

, (1.7)

where({ ˆS_u}u∈[0,∞),P)denotes the continuous-time, simple symmetric random walk onZ^d. One can check that the functionuˆ_t,ξ(x)is the solution of the following Cauchy problem:

_∂

∂tuˆ_t,ξ(x)=uˆ_t,ξ(x)+ξ(x)uˆ_t,ξ(x), ˆ

u_0,ξ(x)=¹0(x) for(t, x)∈(0,∞)×Z^d,

known in the literature as theparabolic Anderson problem. We refer to [4–6] and references therein for the physical motivations behind this problem and for a survey of the main results.

When the potentialξ is i.i.d. with heavy tails like in (1.2) andα > d, the asymptotic properties ast→ ∞of the functionuˆt,ξ(·)were investigated in [8], showing that a very strong form of localization takes place: for larget, the function uˆt,ξ(·)is essentially concentrated on two points almost surely and on a single point in probability. More precisely, for allt >0 andξ∈Ωξ there existzˆ⁽¹⁾_t,ξ,zˆ⁽²⁾_t,ξ∈Z^dsuch that

tlim→∞

ˆ

u_t,ξ(zˆ⁽¹⁾_t,ξ)+ ˆu_t,ξ(zˆ⁽²⁾_t,ξ)

x∈Z^duˆt,ξ(x) =1, P-almost surely, (1.8)

tlim→∞

ˆ u_t,ξ(zˆ⁽¹⁾_t,ξ)

x∈Z^duˆt,ξ(x)=1, inP-probability, (1.9) cf. [8, Theorems 1.1 and 1.2]. The points ˆz⁽¹⁾_t,ξ,zˆ⁽²⁾_t,ξ are typically at superballistic distance(t/logt )^1+q withq = d/(α−d) >0, cf. [8], Remark 6. We point out that localization at one point like in (1.9) cannot holdP-almost surely,

that is, the contribution ofzˆ⁽²⁾_t,ξcannot be removed from (1.8): this is due to the fact thatuˆ_t,ξ(x)is a continuous function oft for every fixedx∈Z^d, as explained in [8], Remark 1.

It is natural to ask if the discrete-time counterpart ofuˆ_t,ξ(·), i.e., the constrained partition functionu_N,ξ(·)defined in (1.5), exhibits similar features. Generally speaking, models built over discrete-time or continuous-time simple random walks are not expected to be very different. However, due to the heavy tail of the potential distribution, the localization points ˆz⁽¹⁾_t,ξ,zˆ⁽²⁾_t,ξ of the continuous-time model grow at a superballistic speed, a feature that is clearly impossible for the discrete-time model, for whichu_N,ξ(x)≡0 for|x|> N. Another interesting question is whether for the discrete model one may have localization at one single pointP-almost surely. Before answering, we need to set up some notation.

We recall thatBN:= {x∈Z^d: |x| ≤N}. It is not difficult to check that the values{pN,ξ(x)}x∈BN are all distinct, forP-a.e.ξ∈Ω_ξ and for allN∈N, because the potential distribution is continuous, cf. (1.2). Therefore we can set

wN,ξ :=arg max

p_N,ξ(x): x∈BN

(1.10)

andP-almost surely wN,ξ is a single point inZ^d: it is the point at whichp_N,ξ(·)attains its maximum. We can now state our first main result.

Theorem 1.3 (One-site localization). We have

Nlim→∞p_N,ξ(wN,ξ)= lim

N→∞

u_N,ξ(wN,ξ)

x∈Z^du_N,ξ(x)=1, P(dξ )-almost surely. (1.11) Furthermore,asN→ ∞we have the following convergence in distribution:

wN,ξ

N ⇒w, whereP(w∈dx)=c_α(1− |x|)^α1_{|x|≤1}dx (1.12)

andc_α:=(

|y|≤1(1− |y|)^αdy)⁻¹.

Recalling the definition (1.6) ofp_N,ξ(x), Theorem1.3shows thatS_N underPN,ξ is localized at the ballistic point wN,ξ≈w·N.

Next we look more closely at the localization site wN,ξ. We introduce two pointsz⁽¹⁾_N,ξ, z⁽²⁾_N,ξ∈Z^d, defined explicitly in terms of the potentialξ, through

z⁽¹⁾_N,ξ :=arg max

1− |x| N+1

ξ(x): x∈BN

,

(1.13) z⁽²⁾_N,ξ :=arg max

1− |x| N+1

ξ(x): x∈BN\ z⁽¹⁾_N,ξ

.

Again, the values of{(1−_N^|x₊^|₁)ξ(x)}x∈BNareP-a.s. distinct, by the continuity of the potential distribution, therefore z_N,ξ⁽¹⁾ andz⁽²⁾_N,ξ areP-a.s. single points inBN. We can now give the discrete-time analogues of (1.8) and (1.9).

Theorem 1.4 (Two-sites localization). The following relations hold:

Nlim→∞

p_N,ξ z_N,ξ⁽¹⁾

+p_N,ξ z⁽²⁾_N,ξ

=1 P(dξ )-almost surely, (1.14)

Nlim→∞p_N,ξ z⁽¹⁾_N,ξ

=1 inP(dξ )-probability. (1.15)

Putting together Theorems1.3and1.4, we obtain the following information on wN,ξ. Corollary 1.5. ForP-a.e.ξ∈Ω_ξ,we havewN,ξ∈ {z⁽¹⁾_N,ξ, z⁽²⁾_N,ξ}for largeN.Furthermore,

Nlim→∞P

wN,ξ =z⁽¹⁾_N,ξ

=1. (1.16)

In Proposition1.6below, we stress that the convergence in (1.15) does not occurP(dξ )-almost surely in dimension d=1, i.e., wN,ξ is not equal toz_N⁽¹⁾for allN large enough. We strongly believe that the latter remains true ford >1.

Proposition 1.6. In dimensiond=1,we have P

wN,ξ=z⁽²⁾_N,ξ for infinitely manyN

=1. (1.17)

The proof of two sites localization given in [8] for the continuous-time model is quite technical and exploits tools from potential theory and spectral analysis. We point out that such tools can be applied also in the discrete-time setting, but they turn out to be unnecessary. Our proof is indeed based on shorter and simpler geometric arguments.

For instance, we exploit the fact that before reaching a site x∈Z^d a discrete-time random walk path must visit a least|x| −1 different sites (=x) and spend at each of them a least one time unit. Of course, this is no longer true for continuous-time random walks.

1.5. Further path properties

Theorem1.3states thatP(dξ )-a.s. the probability measurePN,ξ concentrates on the subset of Ω_S gathering those random walk trajectoriesSsuch thatS_N=wN,ξ. It turns out that this subset can be radically narrowed. In fact, we can introduce a restricted subsetCN,ξ⊆Ω_S of random walk trajectories, defined as follows:

• the trajectories inCN,ξ must reach the site wN,ξ for the first time before timeN, following an injective path, and then must remain at wN,ξ until timeN;

• the length of the injective path until wN,ξdiffers from|wN,ξ|– which is the minimal one – at most for a small error termhN:=(log logN )^2/αN^1−1/αifα >1 andhN:=(logN )^1+2/αifα≤1 (note that in any casehN=o(N ));

• all the sitesx visited by the random walk before reaching wN,ξ must have an associated fieldξ(x)that is strictly smaller thanξ(wN,ξ).

More formally, denoting by τ_x=τ_x(S):=inf{n≥0: S_n=x}the first passage time atx ∈Z^d of a random walk trajectoryS, we set

CN,ξ :=

S∈Ω_S: S_i=S_j∀i < j≤τ_wN,ξ, S_i=wN,ξ∀i∈ {τ_wN,ξ, . . . , N}, ξ(S_i) < ξ(wN,ξ)∀i < τ_wN,ξ, τ_wN,ξ ≤ |wN,ξ| +h_N

. (1.18)

We then have the following result.

Theorem 1.7. ForP-a.e.ξ∈Ω_ξ,we have

Nlim→∞PN,ξ(CN,ξ)=1. (1.19)

Remark 1.8. It is worth stressing that in dimensiond=1the setCN,ξ reduces toa singleN-steps trajectory.In fact, we haveCN,ξ=S^(N,wN,ξ),where we denote byS^(N,x),forx∈BN,the set of trajectoriesS∈Ω_S such that

S_i:=

i·sign(x) for0≤i≤ |x|, x for|x| ≤i≤N.

As stated in Corollary 1.5,for largeN the sitewN,ξ is eitherz⁽¹⁾_N,ξ orz⁽²⁾_N,ξ.Note thatz⁽¹⁾_N,ξ and z⁽²⁾_N,ξ are easily determined,by(1.13).In order to decide whetherwN,ξ =z⁽¹⁾_N,ξ or wN,ξ =z⁽²⁾_N,ξ,by Theorem1.7it is sufficient to compare the explicit contributions of just two trajectories,i.e.,PN,ξ(S^(N,z(1)N,ξ))andPN,ξ(S^(N,z(2)N,ξ)).More precisely, settingκ(i):=P(S1=i)fori∈ {±1,0}(so thatκ=κ(0),cf. (1.1))and

bN,ξ(x):=e^{|x|−1i=1 ξ(isign(x))+(N+1−|x|)ξ(x)}κ

sign(x)_|x|κ(0)^N−|x|,

we havew_N,ξ=z⁽¹⁾_N,ξifb_N,ξ(z⁽¹⁾_N,ξ) > b_N,ξ(z⁽²⁾_N,ξ)andw_N,ξ=z⁽²⁾_N,ξotherwise.Therefore,in dimensiond=1,we have a very explicit characterization of the localization pointwN,ξ.

Remark 1.9. A strong localization result is displayed in[1]for the directed polymer model with a “very heavy tailed”

random environment(α <2).It is shown that,in any dimension,the polymer moves balistically in the hyperplane orthogonal to the deterministic direction.Moreover,for anyδ >0,the polymer of size N remains,with probability 1−e^−cδN(cδ>0),in a cylinder of widthδNaround the trajectory that minimizes the energy.If the inverse temperature βis rescalled byN^1−2/α,the attracting trajectory becomes the minimizer of a variational formula involving both an energetic and an entropic term.

1.6. Organization of the paper The paper is organized as follows.

• In Section2we gather some basic estimates on the field, that will be the main tool of our analysis.

• In Section3we prove Theorem1.4.

• In Section4we prove Theorem1.3.

• In Section5we prove Proposition1.6.

• In Section6we prove Theorem1.7.

• Finally, theAppendicescontain the proofs of some technical results.

In the sequel, the dependence onξ of various quantities, likeHN,ξ, wN,ξ,z⁽¹⁾_N,ξ, etc., will be frequently omitted for short.

2. Asymptotic estimates for the environment

This section is devoted to the analysis of the almost sure asymptotic properties of the random potentialξ. With the exception of Proposition2.5, which plays a fundamental role in our analysis, the proof of the results of this section are obtained with the standard techniques of extreme values theory and are therefore deferred to the AppendicesAandB.

Before starting, we set up some notation. We say that a property of the fieldξdepending onN∈Nholdseventually P-a.s.if forP-a.e.ξ∈Ω_ξ there existsN₀=N₀(ξ ) <∞such that the property holds for allN≥N₀. We recall that

| · |denotes the¹norm onR^dandBN= {x∈Z^d: |x| ≤N}. With some abuse of notation, the cardinality ofBNwill be still denoted by|BN|. Note that|BN| =c_dN^d+O(N^d−1)asN→ ∞, wherec_d=

R^d1_{|x|≤1}dx=2^d/d!.

2.1. Order statistics for the field

The order statistics of the field{ξ(x)}x∈BN is the set of values attained by the field rearranged in decreasing order, and is denoted by

X⁽¹⁾_N > X⁽²⁾_N >· · ·> X⁽_N^|BN|)>1. (2.1)

For simplicity, whent∈ [1,|BN|] is not an integer we still setX^(t)_N :=X_N^(t). For later convenience, we denote by x_N^(k) the point inBN at which the valueX^(k)_N is attained, that isX_N^(k)=ξ(x_N^(k)). We are going to exploit heavily the following almost sure estimates.

Lemma 2.1. For everyε >0,eventuallyP-a.s.

N^d/α

(log logN )^1/α+ε ≤X_N⁽¹⁾≤N^d/α(logN )^1/α+ε. (2.2)

For everyθ >1andε >0,eventuallyP-a.s.

N^d/α

(logN )^θ/α+ε ≤X^{((logN )}_N ^θ)≤ N^d/α

(logN )^θ/α−ε. (2.3)

There exists a constantA >0such that eventuallyP-a.s.

sup

(logN )≤k≤|BN|

k^1/αX_N^(k)

≤AN^d/α. (2.4)

The proof of Lemma2.1is given in AppendixA.2. For completeness, we point out thatX⁽¹⁾_N /(c_dN^d/α)converges in distribution as N → ∞toward the lawμ on(0,∞)withμ((0, x])=exp(−x^−α), called Fréchet law of shape parameterα, as one can easily prove.

Next we give a lower bound on the gapsX^(k)_N −X_N^(k+1)for moderate values ofk.

Proposition 2.2. For everyθ >0there exists a constantγ >0such that eventuallyP-a.s.

inf

1≤k≤(logN )^θ

X^(k)_N −X_N^(k+1)

≥ N^d/α

(logN )^γ. (2.5)

The proof of Proposition2.2is given in AppendixA.3.

2.2. Order statistics for the modified field

An important role is played by the modified field{ψ_N(x)}x∈BN, defined by ψN(x):=

1− |x| N+1

ξ(x). (2.6)

The motivation is the following: for any given pointx∈BN, a random walk trajectory(S0, S1, . . . , SN)that goes tox in the minimal number of steps and sticks inxafterwards has an energetic contribution equal to_|x|−1

i=1 ξ(Si)+(N+ 1)ψN(x)(recall (1.3)).

The order statistics of the modified field{ψ_N(x)}x∈BN will be denoted by Z⁽¹⁾_N > Z_N⁽²⁾>· · ·> Z^|B_N^N|,

and we letz^(k)_N be the point inBNat whichψ_NattainsZ^(k)_N , that isψ_N(z^(k)_N )=Z_N^(k). A simple but important observation is that Z_N^(k) is increasing inN, for every fixedk∈N, since ψ_N(x) is increasing inN for fixed x. Also note that Z^(k)_N ≤X_N^(k), becauseψN(x)≤ξ(x).

Our attention will be mainly devoted toZ_N⁽¹⁾andZ⁽²⁾_N , whose almost sure asymptotic behaviors are analogous to that ofX_N⁽¹⁾, cf. (2.2).

Lemma 2.3. For everyε >0,eventuallyP-a.s.

N^d/α

(log logN )^1/α+ε ≤Z_N⁽²⁾≤Z_N⁽¹⁾≤N^d/α(logN )^1/α+ε. (2.7) The proof is given in AppendixB.2. Note that only the first inequality needs to be proved, thanks to (2.2) and to the fact that, plainly,Z⁽²⁾_N ≤Z_N⁽¹⁾≤X_N⁽¹⁾.

Next we focus on the gaps betweenZ_N⁽¹⁾, Z_N⁽²⁾ andZ⁽³⁾_N . The main technical tool is given by the following easy estimates, proved in AppendixB.1.

Lemma 2.4. There is a constantcsuch that for allN∈Nandδ∈(0,1) P

Z⁽²⁾_N > (1−δ)Z⁽¹⁾_N

≤cδ, P

Z_N⁽³⁾> (1−δ)Z⁽¹⁾_N

≤cδ². (2.8)

As a consequence, we have the following result, which will be crucial in the sequel.

Proposition 2.5. For everyd andα,there existsβ∈(1,∞)such that

Z_N⁽¹⁾−Z_N⁽³⁾≥ N^d/α

(logN )^β, eventuallyP-a.s. (2.9)

Although we do not use this fact explicitly, it is worth stressing that the gapZ_N⁽¹⁾−Z_N⁽²⁾can be as small asN^d/α−1 (up to logarithmic corrections), hence much smaller than the right hand side of (2.9), cf. AppendixB.3. This is the reason behind the fact that localization at the two points {z⁽¹⁾_N,ξ, z⁽²⁾_N,ξ}can be proved quite directly, cf. Section 3, whereas localization at a single point wN,ξ ∈ {z⁽¹⁾_N,ξ, z⁽²⁾_N,ξ}is harder to obtain, cf. Section4. Furthermore, one may have wN,ξ=z⁽¹⁾_N,ξ precisely when the gapZ_N⁽¹⁾−Z⁽²⁾_N is small, cf. Section5.

Proof of Proposition2.5. Forr∈(0,1)(that will be fixed later), we setN_k:= e^kr, fork∈N. By the second relation in (2.8), forγ >0 (to be fixed later) we have

k∈N

P

Z_N⁽¹⁾

k −Z⁽³⁾_N

k ≤ 1

(logNk)^γZ_N⁽¹⁾

k

≤c₁

k∈N

1

(logNk)^2γ ≤(const.)

k∈N

1 k^2rγ <∞,

provided 2rγ >1. Therefore, by the Borel–Cantelli lemma and (2.7), eventually (ink)P-a.s.

Z_N⁽¹⁾

k−Z_N⁽³⁾

k ≥ (Nk)^d/α

(logN_k)^γ+1. (2.10)

Now for a genericN∈N, letk∈Nbe such thatN_k−1≤N < N_k. We can write Z_N⁽¹⁾−Z_N⁽³⁾=

Z_N⁽¹⁾−Z_N⁽¹⁾

k

+ Z⁽¹⁾_N

k −Z_N⁽³⁾

k

+ Z⁽³⁾_N

k −Z_N⁽³⁾ .

We already observed thatZ_N^(k)is increasing inN, therefore the third term in the right hand side is non-negative and can be neglected. From (2.10) we then get for largeN

Z_N⁽¹⁾−Z_N⁽³⁾≥ (N_k)^d/α (logNk)^γ+1−

Z⁽¹⁾_N

k −Z_N⁽¹⁾

≥ N^d/α 2(logN )^γ+1−

Z⁽¹⁾_N

k −Z_N⁽¹⁾

(2.11) becauseNk≥N andNk≤2N for largeN (note thatNk/Nk−1→1 ask→ ∞).

It remains to estimateZ⁽¹⁾_N

k −Z_N⁽¹⁾. Observe thatZ_n⁽¹⁾=ψn(z⁽¹⁾_n )≥ψn(z⁽¹⁾_n+1), becauseZ⁽¹⁾_n is the maximum ofψn. Therefore we obtain the estimate

Z_n⁽¹⁾₊₁−Z_n⁽¹⁾=ψ_n+1 z⁽¹⁾_n+1

−ψ_n z⁽¹⁾_n

≤ψ_n+1 z⁽¹⁾_n+1

−ψ_n z⁽¹⁾_n+1

= |z⁽¹⁾_n+1|ξ(z_n⁽¹⁾₊₁)

(n+1)(n+2)≤ξ(z⁽¹⁾_n+1) n which yields

Z_N⁽¹⁾

k−Z_N⁽¹⁾=

Nk−1 n=N

Z_n⁽¹⁾₊₁−Z⁽¹⁾_n

≤ N_k−N_k−1 N_k−1 ξ

z_N⁽¹⁾

k

≤N_k−N_k−1 N_k−1 X_N⁽¹⁾

k. (2.12)

Observe that ask→ ∞ e^kr−e^(k−1)r

e^(k−1)r =e^{kr−(k−1)r}−1= r k^1−r

1+o(1)

. (2.13)

SinceN≤N_k= e^kr, it comes thatk≥(logN )^1/r and therefore (2.13) allows to write for largeN Nk−Nk−1

N_k−1 ≤ 1

(logN )^1/r−1.

Looking back at (2.11) and (2.12), by (2.2) we then have eventuallyP-a.s.

Z⁽¹⁾_N −Z_N⁽³⁾≥ N^d/α

2(logN )^γ+1− N^d/α

(logN )^{1/r−1/α−2}. (2.14)

The second term in the right hand side of (2.14) can be neglected provided the parametersr∈(0,1)andγ∈(0,∞) fulfill the condition 1/r−1/α−2> γ +1. We recall that we also have to obey the condition 2rγ >1. Therefore, for a fixed value of r, the set of allowed values for γ is the interval (_2r¹,¹_r −_α¹ −3), which is non-empty if r is small enough. This shows that the two conditions onr, γ can indeed be satisfied together (a possible choice is, e.g., r=_6(3α^α₊₁₎ andγ=^4(3α_α⁺¹⁾). Settingβ:=γ+1, it then follows from (2.14) that equation (2.9) holds true.

3. Almost sure localization at two points

In this section we prove Theorem1.4. We first set up some notation and give some preliminary estimates.

3.1. Prelude

We recall thatz⁽¹⁾_N andz⁽²⁾_N are the two sites inBN at which the modified potentialψ_N, cf. (2.6), attains its two largest valuesZ_N⁽¹⁾=ψ_N(z⁽¹⁾_N )andZ_N⁽²⁾=ψ_N(z⁽²⁾_N ).

It is convenient to define J₁, J₂∈ {1, . . . ,|BN|}as the ranks (in the order statistics) of the two sites where the modified potential reaches its maximum and its second maximum, respectively. Thus,

z⁽¹⁾_N =x_N^(J1), z⁽²⁾_N =x_N^(J2), (3.1)

where we recall thatx_N^(k)is the point inBNat which the potentialξ attains itskth largest value, i.e.,X^(k)_N =ξ(x_N^(k)), cf. Section2.1. We stress thatJ₁andJ₂ are functions ofN andξ, although we do not indicate this explicitly. An immediate consequence of Lemma2.3and relation (2.3) is the following

Corollary 3.1. For everyd,α,ε >0,eventuallyP-a.s.

max{J₁, J₂} ≤(logN )^1+ε. (3.2)

Next we define the local time_N(x)of a random walk trajectoryS∈Ω_Sby

_N(x)=_N(x, S)= N i=1

1_{Si=x}, (3.3)

so that the HamiltonianH_N(S), cf. (1.3), can be rewritten as HN(S)=

x∈BN

N(x)ξ(x). (3.4)

We also associate to every trajectorySthe quantity βN(S):=min

k≥1: N

x_N^(k)

>0

. (3.5)

In words,x_N^(βN(S)) is the site inBN which maximizes the potentialξ among those visited by the trajectoryS before timeN. Finally, we introduce the basic events

A1,N :=

S∈Ω_S: β_N(S)=J₁

, A2,N :=

S∈Ω_S: β_N(S)=J₂

. (3.6)

In words, the eventAi,Nconsists of the random walk trajectoriesSthat before timeN visit the sitez⁽ⁱ⁾_N (recall(3.1)) and do not visit any other sitex withξ(x) > ξ(z⁽ⁱ⁾_N).

It turns out that the localization ofS_Nat the pointz⁽ⁱ⁾_N is implied by the eventAi,N, i.e., for bothi=1,2 we have

Nlim→∞PN,ξ

Ai,N, S_N=z_N⁽ⁱ⁾

=0, P(dξ )-almost surely. (3.7)

The proof is simple. Denoting byτ_N,ithelastpassage time of the random walk inz⁽ⁱ⁾_N before timeN, that is τ_N,i:=max

n≤N:S_n=z⁽ⁱ⁾_N , we can write, recalling (1.3),

PN,ξ

Ai,N, S_N=z⁽ⁱ⁾_N

=

N−1 r=0

E[e^HN(S)1_Ai,N¹_{τN,i=r}] UN,ξ

. (3.8)

We stress that the sum stops atr=N−1, because we are on the eventS_N=z⁽ⁱ⁾_N. Furthermore, on the eventAi,N∩ {τ_N,i=r} we have S_n∈ {/ x_N⁽¹⁾, . . . , x_N^(Ji)}for all n∈ {r+1, . . . , N}(we recall that z⁽ⁱ⁾_N =x_N^(Ji)). By the Markov property, we can then bound the numerator in the right hand side of (3.8) by

E

e^HN(S)1_Ai,N1_{τN,i=r}

≤E e^Hr(S)1_{

Sr=z⁽ⁱ⁾_N}

B_N^(N,i)_−r,

whereB_l^(N,i):=E

x^{(Ji )}_N

e^Hl(S)1

{S_n∈{/x⁽¹⁾_N,...,x^{(Ji )}_N }∀n=1,...,l}

. (3.9)

Analogously, for the denominator in the right hand side of (3.8), recalling (1.1), we have U_N,ξ ≥E

e^HN(S)1

{Sn=x^{(Ji )}_N ,∀n∈{r,...,N}}

=E e^Hr(S)1

{Sr=x_N^{(Ji )}}

κ^N−re^{(N−r)X(Ji )N} .

Plainly,B_l^(N,i)≤exp(lX_N^(Ji+1)), therefore we can write PN,ξ

Ai,N, S_N=z⁽ⁱ⁾_N

≤

N−1 r=0

e^{−(N−r)(X(Ji )N +logκ)}B_N^(N,i)_−r = N

l=1

e^{−l(X(Ji )N +logκ)}B_l^(N,i)

≤ ∞ l=1

e^{−l(X(Ji )N −X(JiN+1)−logκ)}= e^{−(X(Ji )N −X(JiN+1)−logκ)} 1−e^{−(XN(Ji )−XN(Ji+1)−logκ)}

. (3.10)

From Corollary 3.1and Proposition2.2 it follows that P(dξ )-almost surelyX^(J_Nⁱ⁾−X_N^(Ji+1)→ +∞ as N → ∞, therefore (3.7) is proved.

3.2. Proof of(1.14) Let us set, fori=1,2,

Wi,N :=

S∈ΩS: βN(S)=Ji, SN=z⁽ⁱ⁾_N

=

S∈ΩS: SN=z⁽ⁱ⁾_N, N(x)=0∀x∈BNsuch thatξ(x) > ξ z⁽ⁱ⁾_N

. (3.11)