2 Proof of Theorem B

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El e c t ro nic J

ou o

f Pr

ob a bi l i t y

Electron. J. Probab.18(2013), no. 97, 1–20.

ISSN:1083-6489 DOI:10.1214/EJP.v18-2231

An almost sure CLT for stretched polymers

Dmitry Ioffe

^∗

Yvan Velenik

^†

Abstract

We prove an almost sure central limit theorem (CLT) for spatial extension of stretched (meaning subject to a non-zero pulling force) polymers at very weak disorder in all dimensionsd+ 1≥4.

Keywords:Polymers; random walk representation; random environment; weak disorder; CLT.

AMS MSC 2010:60F05; 60K35; 82B41; 82B44; 82D60.

Submitted to EJP on August 9, 2012, final version accepted on November 2, 2013.

SupersedesarXiv:1207.5687v2.

1 Introduction and Results

Directed polymers in random media were introduced in [7] as an effective model of Ising interfaces in systems with random impurities. The precise mathematical formula- tion appeared in the seminal paper [9], which triggered a wave of subsequent investi- gations. The model of directed polymers can be described as follows. Letη = (η_k)_0≤k≤n be a nearest-neighbour path onZ^dstarting at0, and letγ= (γk)_0≤k≤nwithγk = (k, ηk) be the corresponding directed path inZ^d+1^{. Let also}{V(x)}_x∈Zd+1 be a collection of i.i.d. random variables with finite exponential moments, whose joint law is denoted by P. One is then interested in the behaviour of the pathγ under the random probability measure

µ^ω_n(γ) = (Z_n;β^ω )⁻¹exp −β

n

X

k=1

V(γk)

(2d)⁻ⁿ,

whereβ ≥0 is the inverse temperature. The behaviour of the pathγis closely related to the behaviour of the partition functionZ_n;β^ω . Namely, one distinguishes between two regimes: the weak disorder regime, in whichlim_n→∞Z_n;β^ω /E(Z_n;β^ω )>0,P-a.s., and the strong disorder regime, in which this limit is zero. It is known [2] that there is a sharp transition between these two regimes at an inverse temperatureβcwhich is non-trivial whend ≥3. In the weak disorder regime (β < β_c), the path γ behaves diffusively, in thatγ_n satisfies a CLT. Diffusivity at sufficiently small values ofβ was first established in [9]; this was extended to an almost-sure CLT in [1]; a CLT (in probability) valid in the whole weak disorder regime was then obtained in [2].

∗Technion E-mail:ieioffe@ie.technion.ac.il

†Université de Genève E-mail:Yvan.Velenik@unige.ch

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In dimensionsd≥3the sequenceZ_n;β^ω /E(Z_n;β^ω )is bounded inL2 for all sufficiently small values of β. In such a situation local limit versions of the CLT, which hold in probability, were established in [16, 18].

In the case of directed polymers the disorder is always strong in dimensions d = 1,2 [3, 14] and at sufficiently low temperatures. Concerning the (nondiffusive) behaviour in the strong disorder regime, we refer the reader to [4] and references therein.

In this work, we consider diffusive behaviour in dimensionsd+ 1≥4for the related models ofstretched polymers. The choice of notationd+ 1indicates that stretched polymers onZ^d+1should be compared with directed polymers inddimensions. However, a stretched pathγ can be any nearest-neighbour path onZ^d+1, which is permitted to bend and to return to particular vertices an arbitrary number of times. The disorder is modelled by a collection{V(x)}_x∈_Zd+1 of i.i.d. non-negative random variables. Each visit of the path to a vertexxexerts the pricee^−βV^(x). Thestretch is introduced in one of the following two natural ways:

• The path γ starts at 0 and ends at a hyperplane at distance n from 0 and has arbitrary length. This is a model of crossing random walks in random potentials.

In dimension d+ 1 = 2, it presumably provides a better approximation to Ising interfaces in the presence of random impurities.

• The pathγhas a fixed lengthn, but it is subject to a drift, which can be interpreted physically as the effect of a force acting on the polymer’s free end.

The precise model is described below. At this stage let us remark that models of stretched polymers have a richer morphology than models of directed polymers. Even the issue of ballistic behaviour for annealed models is non-trivial [10, 8, 13]. The issue of ballistic behaviour in the quenched case is still not resolved completely, and, in order to ensure ballisticity one needs to assume that the random potential V is strictly positive in the crossing case, and that the applied drift is sufficiently large in the fixed length case. Both conditions are designed to ensure a somewhat massive nature of the model.

As in the directed case, the disorder is always strong [21] in low dimensionsd+ 1 = 2,3or at sufficiently low temperatures.

In the case of higher dimensionsd+1≥4, the existence of weak disorder on the level of equality between quenched and annealed free energies was established in [6, 20].

The case of high temperature discrete Wiener sausage with drift was addressed in [17].

In the crossing case, a CLT in probability was established in [11] in all dimensions d+ 1≥4at sufficiently high temperatures.

The aim of the present paper is to establish an almost-sure CLT for the endpoint of the fixed-length version of the model of stretched polymers with non-zero drifts, also at sufficiently high temperatures and in all dimensionsd+ 1≥4.

1.1 Class of Models

Polymers. For the purpose of this paper, a polymer γ = (γ₀, . . . , γ_n) is a nearest- neighbour trajectory on the integer latticeZ^d+1. Unless stressed otherwise, γ₀ is always placed at the origin. The length of the polymer is|γ|=^∆nand its spatial extension isX(γ)=^∆γ_n−γ₀. In the most general case, neither the length nor the spatial extension are fixed.

Random Environment. The random environment is a collection{V(x)}_x∈_Zd+1 of non- degenerate non-negative i.i.d. random variables which are normalised by0∈supp(V). There is no moment assumptions onV. The case of traps,p∞

=∆ P(V =∞)>0, is not

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excluded, but then we shall assume thatp_∞is small enough. In particular, we shall assume thatP-a.s. there is an infinite connected clusterCl_∞(V)of the set{x:V(x)<∞}

inZ^d+1. In fact, we shall assume more: GivenR^d+13h6= 0and a numberδ∈(0,^√¹

d+1), define the positive cone

Y_δ^h=^∆

x∈R^d+1 : x·h≥δ|x| |h| . (1.1) By construction, the cones Y_δ^h always contain at least one lattice direction ±ei, i = 1, . . . , d+1. We assume that it is possible to chooseδin such a fashion that, for anyh, the intersectionCl^h,δ_∞ (V)= Cl^∆ _∞(V)∩ Y_δ^hcontains (P-a.s.) an infinite connected component.

For the rest of the paper, we fix such aδ∈ (0,^√_d+1¹ )and use the reduced notationY^h andCl^h_∞(V)for the corresponding cones (1.1) and percolation clusters.

Weights and Path Measures. The reference measure p(γ) = (2(d^∆ + 1))^−|γ| is given by simple random walk weights. The polymer weights we are going to consider are quantified by two parameters: the inverse temperatureβ ≥0and the external pulling forceh∈R^d+1^.

The random quenched weights are given by q_h,β^ω (γ)= exp^∆ n

h·X(γ)−β

|γ|

X

1

V(γi)o

p(γ). (1.2)

The corresponding deterministic annealed weights are given by

qh,β(γ)^∆=Eq_h,β^ω (γ) = exp{h·X(γ)−Φβ(γ)}p(γ), (1.3) whereΦβ(γ)=^∆P

xφβ `γ(x)

, with`γ(x)denoting the local time (number of visits) ofγ atx, and

φβ(`)=^∆−logEe^−β`V. (1.4)

Note that the annealed potential is positive, non-decreasing and attractive, in the sense that

0< φ_β(`)≤φ_β(`+m)≤φ_β(`) +φ_β(m), ∀`, m∈N. (1.5) In the sequel, we shall drop the indexβfrom the notation, and we shall drop the index hwhenever it equals zero. With this convention, the quenched partition functions are defined by

Q^ω_n(x)=^∆ X

X(γ)=x

|γ|=n

q^ω(γ), Q^ω_n(h)=^∆ X

|γ|=n

q^ω_h(γ) =X

x

e^h·xQ^ω_n(x), (1.6)

and we useQn(x) =^∆ EQ^ω_n(x)and Qn(h) ^∆= EQ^ω_n(h) to denote their annealed counterparts.

Finally, we define the corresponding quenched and annealed path measures by Q^ωn,h(γ)=^∆1{|γ|=n}

q_h^ω(γ)

Q^ω_n(h) ^and Q^n,h(γ)=^∆1{|γ|=n}

q_h(γ)

Qn(h). (1.7) Very Weak Disorder. The notion of very weak disorder is technical and it depends on the strength |h| of the pulling force , dimension d ≥3 and the distribution of V. By Lemma 2.1 below, there exists a functionζd on(0,∞)such that a certain L²^-estimate (2.4) holds ifφβ(1)< ζd(|h|).

Definition 1.1. The model of stretched polymers is in the regime of very weak disorder ifd≥3and

φ_β(1)< ζ_d(|h|). (1.8)

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1.2 The Result

Fixh6= 0. Then [19, 5, 10]

λ=λ(β, h)= lim^∆

n→∞

1

nlogQ_n(h)∈(0,∞), (1.9) for all sufficiently smallβ. The following two quantities play a central role in our limit theorems:

v=v(h, β)=^∆∇λ(h), Σ= Hess[λ](h).^∆

Ifβis sufficiently small thenv6= 0and the matrixΣis positive definite and, moreover,v andΣare the limiting spatial extension and, respectively, the diffusivity matrix for the annealed model. (Sections 4.1,4.2 in [10]). In Subsection 2.1 we recall further relevant facts about the annealed model.

Theorem A. Fixh6= 0. Then, in the regime of very weak disorder, the following holds P-a.s. on the event{0∈Cl_∞(V)}:

• The limit

n→∞lim Q^ω_n(h)

Qn(h) ^(1.10)

exists and is a strictly positive, square-integrable random variable.

• There exists a sequence{n}withlimn = 0, such that X

n

Q^ωn,h

X(γ) n −v

> n

<∞. (1.11)

• For everyα∈R^d+1^,

n→∞lim Q^ωn,h

expiα

√n(X(γ)−nv)

= exp

−¹₂Σα·α . (1.12) We would like to stress that, in contrast to the case of directed polymers [2], our CLT does not pertain to the whole of the weak disorder region. The procedure of first fixingh6= 0and then going toβ >0sufficiently small is essential. Furthermore, even in the regime we are working with, (1.12) falls short of the local CLT form of results as developed for directed polymers in [18]. These and related issues remain open in the context of stretched polymers.

Few remarks on the history of the problem: Flury [6] had established that under the conditions of Theorem A (and some additional moment assumptions of the potentialV)

n→∞lim 1

nlogQ^ω_n(h)

Q_n(h) = 0 (1.13)

for on-axis exterior forcesh. (1.13) was then extended to arbitrary directionsh∈R^d+1 by Zygouras [20]. In [6], the analysis was carried out directly in the canonical ensemble of polymers with fixed lengthn. In [20], the author derives results for the conjugate ensemble of the so-called crossing random walks.

Large deviations (LD) under bothQ^n,h^andQ^ωn,hwere investigated in [19, 5]. The results therein imply that, under the conditions of Theorem A, the model is ballistic in the sense that the value of the quenched rate function at zero is strictly positive. However, [19, 5] do not imply a law of large numbers (LLN) even in the annealed case. In particular, these works do not contain information on the strict convexity of the corresponding rate functions. The issue of strict convexity for the annealed rate functions was set- tled in [10]. Therefore, (1.11) is a direct consequence of (1.13) and of the analysis of annealed canonical measures in [10].

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The main new results of this work are (1.10) and (1.12). A version of Theorem A for the ensemble of crossing random walks appears in [11]. The length of crossing random walks is not fixed (only suppressed by an additional positive mass), and they are required to have their second endpoint on a distant hyperplane. In this way, crossing random walks in random potential are much more “martingale”-like than canonical random walks. Moreover, the canonical constraint of fixed length does not facilitate computations, to say the least. Finally, the CLT of [11] was only established in probability and notP-a.s. Thus, although the techniques developed in [11] are useful here, they certainly do not imply the claims of Theorem A, and an alternative approach was required.

1.3 Irreducible Decomposition, Basic Ensembles and Basic Partition Func- tions

A polymerγ= (γ0, . . . , γn)is said to be cone-confined if γ⊂ γ0+Y^h

∩ γn− Y^h

. (1.14)

A cone-confined polymer which cannot be represented as the concatenation of two (non- singleton) cone-confined polymers is said to be irreducible. We denote by T(x) the collection of all cone-confined paths leading from0 tox, and byF(x) ⊂ T(x)the set of irreducible cone-confined paths. In the sequel we shall refer toF(x)andT(x)as to basic ensembles. The basic partition functions are defined by

t^ω_x,n= e^∆ ^−λn X

γ∈T(x)

1{|γ|=n}q^ω_h(γ) and f_x,n^ω = e^∆ ^−λn X

γ∈F(x)

1{|γ|=n}q_h^ω(γ). (1.15)

We also set, accordingly,t^ω_n =^∆P

xt^ω_x,nandf_n^ω=^∆P

xf_x,n^ω . The annealed counterparts of all these quantities are denoted bytx,n

=∆ Et^ω_x,n,fx,n

=∆ Ef_x,n^ω ,tn

=∆ Et^ω_n and fn

=∆ Ef_n^ω. As shown in Section 3.6 of [10], the collection{fx,n}forms a probability distribution,

X

n

X

x

f_x,n=X

n

f_n= 1,

with exponentially decaying tails:

X

m≥n

f_m=^∆ X

m≥n

X

x

f_x,m≤e^−νn, (1.16)

whereν=ν(β, h)→ ∞asβ becomes large, andinf_β≥0ν(β, h)>0, for allh6= 0.

Remark 1.2. Since by definition polymers are nearest neighbour paths, it always holds thatt_x,n=t_x,n1{|x|≤n}.

As in [11, Subsections 2.7 and 3.5], the following statement about basic ensembles implies the claims (1.10) and (1.12) of Theorem A:

Theorem B. Fixh6= 0. Then, in the regime of very weak disorder, the following holds P-a.s. on the eventn

0∈Cl^h_∞(V)o :

• The limit

s^ω= lim^∆

n→∞

t^ω_n

t_n ^(1.17)

exists and is a strictly positive, square-integrable random variable.

• For everyα∈R^d+1^,

n→∞lim 1 t^ω_n

X

x

expiα

√n ·(x−nv) t^ω_x,n= exp

−¹₂Σα·α . (1.18) For the rest of the paper, we shall focus on the proof of Theorem B.

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2 Proof of Theorem B

To facilitate the exposition, we shall consider the case of on-axis external force h = he1. The proof, however, readily applies for any non-zero h ∈ R^d+1. By lattice symmetries, the mean displacementv =∇λ(h)lies along the directione1;v =ve1. As it was already mentioned in the beginning of Subsection 1.2,v6= 0wheneverβis small enough. We proceed assuming that both the drift and the speed are positiveh, v >0. 2.1 Three Main Inputs

The reduction to basic ensembles constitutes the central step of the Ornstein-Zernike theory. We rely on three facts: The first is the refined description of the annealed phase in the ballistic regime (which, in our regime, will always correspond to first fixingh6= 0 and then choosingβ >0small enough). Below, we shall summarize the required results from [10, 12]. The second is anL2-type estimate on overlaps which holds for allβsuf- ficiently small, and which could be understood as quantifying the notion of very weak disorder we employ here. The third is a maximal inequality for the so-called mixin- gales, due to McLeish. Unlike directed polymers, stretched polymers do not possess natural martingale structures, and McLeish’s result happens to provide a convenient alternative framework.

Ornstein-Zernike theory of annealed models. Annealed asymptotics of t_n in the ballistic regime are not related to the strength of disorder and hold for all values of β≥0and appropriately large driftsh. In particular, for eachh6= 0fixed, the annealed model is ballistic for all sufficiently smallβ. We refer to [10, Sections 4.1 and 4.2] and to [12, Section 4.2] for the proof of the following: Fixh6= 0; then, for allβ > 0 small enough, λ(h) > 0, ∇λ(h) 6= 0and Hess[λ](h)is positive definite. Furthermore, there exist a small complex neighbourhood U ⊂ C^d+1 of the origin, an analytic function µ (withµ(0) = 0) onU and a non-vanishing analytic functionκ6= 0onU such that:

n→∞lim e^−nµ(z)t_n(z)= lim^∆

n→∞e^−nµ(z)X

x

t_x,ne^z·x= 1

κ(z), (2.1)

uniformly exponentially fast onU. Note [12] (Section 4.2) thatλ(h+z) =λ(h) +µ(z)for realz, and thusv=∇λ(h) =∇µ(0)andΣ = Hess[λ](h) = Hess[µ](0).

The annealed model satisfies a local LD upper bound: There exists c =c(β, h)> 0 such that, for allx∈ Y^h,

tx,n≤ 1 c√

n^d+1expn

−c|x−nv|² n

o

. (2.2)

In view of Remark 1.2 the above bound is trivial whenever|x|> n.

Finally, it is a straightforward consequence of (2.1) that the following annealed CLT holds:

Sn

√α n

∆

=X

x

tx,nexp i α

√n·(x−nv) = 1 κ(0)exp

−¹₂Σα·α 1 +O(n^−1/2)

, (2.3)

with the second asymptotic equality holding uniformly inαon compact subsets ofR^d+1^. An L²^-estimate. Fix an external force h 6= 0. We continue to employ notation v = v(h, β). For a subsetA ⊆ Z^d+1^{, let} Abe the σ-algebra generated by{V(x)}_x∈A. We shall call suchσ-algebras cylindrical.

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Lemma 2.1. For any dimensiond≥3there exist a positive non-decreasing functionζ_d on(0,∞)and a numberρ <1/12such that the following holds: Ifφ_β(1)< ζ_d(|h|), then there exist constantsc1, c2<∞such that the random weights (1.15)satisfy:

E

t^ω_x,`t^ω_x0,`E(f_y,m^θ^x^ω−fy,m| A)E(f_y^θ^x0,m⁰^ω⁰−fy⁰,m⁰| A)

≤ c1e^−c²^(m+m⁰⁾

`^d+1−ρ expn

−c2

|x−x⁰|+|x−`v|²

` +|x⁰−`v|²

` o

,

(2.4)

for allx, x⁰, m, m⁰, y, y⁰, ànd all cylindricalσ-algebrasAsuch that botht^ω_x,àndt^ω_x0,àre A-measurable.

Remark 2.2. The above bound is non-trivial only if both|x|,|x⁰| ≤`(Remark 1.2). Also, there is nothing sacred about the conditionρ <1/12. We just needρto be sufficiently small. In fact,(2.4)holds withρ= 0, although a proof of such statement would be a bit more involved.

In spite of its technical appearance, (2.4) has a transparent intuitive meaning: For ρ = 0, the expressions on the right-hand side are just local limit bounds for a couple of independent annealed polymers with exponential penalty for disagreement at their end-points. The irreducible terms have exponential decay. In the very weak disorder regime, the interaction between polymers does not destroy these asymptotics. The proof of Lemma 2.1 is relegated to the concluding Section 4.

McLeish’s Maximal Inequality. LetZ₁, Z₂, . . .be a sequence of zero-mean, square- integrable random variables. Let also{A_k}^∞_−∞ be a filtration ofσ-algebras. Suppose that we have chosen >0and numbersψ1, ψ2, . . .in such a way that

E

E(Z`| A_`−k)²

≤ ψ²_`

(1 +k)¹⁺ ^and E

Z`−E(Z`| A`+k)2

≤ ψ_`²

(1 +k)¹⁺ ^(2.5) for all ` = 1,2, . . .andk ≥0. Then [15] there existsK =K() <∞such that, for all n₁≤n₂,

Eh max

n1≤r≤n2

X^r

n₁

Z`

²i

≤K

n₂

X

n₁

ψ_`². (2.6)

Remark 2.3. In particular, ifP

`ψ_`²<∞, thenP

`Z_`convergesP-a.s. and inL2. In the sequel, we shall always work with the following filtration{Am}. Recall that we are discussing on-axis positive driftsh=he₁ which, for smallβ, give rise to on-axis limiting spatial extensionv=ve₁withv >0. At this stage, define the hyperplanesH⁻_m and the correspondingσ-algebrasAmas

H⁻_m=

x∈Z^d+1 : x·e1≤m|v| and Am=σ

V(x) : x∈ H⁻_m . (2.7) Notation for asymptotic relations. The following notation is convenient, and we shall use it throughout the text: Given a (countable) set of indicesI and two positive sequences{aα, bα}_α∈I, we say that aα . bα if there exists a constant c > 0such that aα ≤ cbα for allα∈ I . We shall use aα ∼= bα if bothaα . bα and aα & bα hold. For instance, for any >0fixed,

e^−c³^k²^/`

`^(1+)/2 . 1

(1 +k)¹⁺, (2.8)

where the index setIis the set of pairs of integers(k, `)withk≥0and` >0.

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Structure of upper bounds. Our upper bounds are based on (2.8), (2.4) (applied withρ=/2) and on (2.6). Recall thatρ <1/12, and hence <1/6.

In the sequel, we shall repeatedly derive variance bounds on quantities of the type P

`≤nZ_`⁽ⁿ⁾. The most general form ofZ_`⁽ⁿ⁾we shall consider is Z_`⁽ⁿ⁾=X

x

t^ω_x,`X

y,m

a⁽ⁿ⁾_x,`(y, m) f_y,m^θ^x^ω−f_y,m

, (2.9)

where

a⁽ⁿ⁾_x,`(y, m) are arrays of real or complex numbers. Assume that there exists another family of (non-negative) arrays

ˆ

a⁽ⁿ⁾_x,` and a numberν >0such that e^−c²^m X

|y|≤m

a⁽ⁿ⁾_x,`(y, m)

.e^−νmaˆ⁽ⁿ⁾_x,`, (2.10) where the constantc2is inherited from (2.4).

Lemma 2.4. Set= ^ρ₂, whereρis the power which shows up in (2.4). Under assump- tion (2.10)

E

E(Z_`⁽ⁿ⁾| A`−k)2

. 1

`^d+1−/2 X

x∈H⁻_`−k

e^−c²^|x−`v|

2

` ˆa⁽ⁿ⁾_x,`2

, (2.11)

and E

Z_`⁽ⁿ⁾−E(Z_`⁽ⁿ⁾| A`+k)2

. 1

`^d+1−/2 X

x

e^−c²^|x−`v|

2

` −νd`+k(x) ˆa⁽ⁿ⁾_x,`2

. (2.12)

Above we introduced a provisional notationdr(x) = (r^∆ |v| −e1·x)∨0for the distance fromxtoH⁺_r =Z^d+1\ H⁻_r.

Proof. SinceE

t^ω_x,` f_y,m^θ^x^ω−f_y,m

| A_`−k

= 0wheverx6∈ H⁻_`−k, E(Z_`⁽ⁿ⁾| A_`−k) = X

x∈H⁻_`−k

t^ω_x,`X

y,m

a⁽ⁿ⁾_x,`(y, m)E(f_y,m^θ^x^ω−fy,m| A_`−k). (2.13)

Taking the expectation of the square of the latter expression and, for each x, x⁰, fac- torizing replicas using|ab| ≤ ^a²^+b₂ ², one derives the first inequality (2.11) directly from (2.4) and (2.10).

Next,

Z_`⁽ⁿ⁾−E(Z_`⁽ⁿ⁾| A`+k) = X

x∈H⁺_`+k

t^ω_x,`X

y,m

a⁽ⁿ⁾_x,`(y, m) f_y,m^θ^x^ω−fy,m

+ X

x∈H⁻_`+k

t^ω_x,` X

z∈H⁺_`+k

X

m

a⁽ⁿ⁾_x,`(z−x, m) f_z−x,m^θ^x^ω −E(f_z−x,m^θ^x^ω | A`+k)

. (2.14)

For anyx∈ H⁺_`+k,d`+k(x) = 0, and the first term in (2.14) has exactly the same structure as the right-hand side of (2.13). On the other hand, ifx∈ H⁻_`+k andz∈ H⁺_`+k, then, in view of Remark 1.2,f_z−x,m^θ^x^ω can be different from zero only ifm≥d`+k(x)and|z−x| ≤ m. Therefore, (2.12) is also a direct consequence of (2.4) and (2.10).

The following is a useful corollary:

Lemma 2.5. Ifˆa⁽ⁿ⁾_x,` .ˆa⁽ⁿ⁾_` , then the bounds (2.11)and (2.12)reduce to E

E(Z_`⁽ⁿ⁾| A`−k)² , E

Z_`⁽ⁿ⁾−E(Z_`⁽ⁿ⁾| A`+k)²

. ˆa⁽ⁿ⁾_` ² 1

`^d/2−(1 +k)¹⁺. (2.15)

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Proof. Consider first the right-hand side of (2.11). Since P

x∈H⁻_`−ke^−c²^|x−`v|

2

` . `^d+1² , the non-trivial part is to check (2.15) for large values ofk. In the latter case, we may assume that|x−v`| > ^k|v|₂ for allx∈ H_`−k⁻ . Consequently, the sum on the right-hand side of (2.11) is bounded above by

X

x∈H⁻_`−k

e^−c²^|x−v`|²^/`. Z

|y|>^k|v|₂

e^−c²^|y|²^/`dy

=∼

Z ∞

k|v|

2

r^de^−c²^r²^/`dr.`^(d+1)/2e^−c³^k²^/`. `^d/2+1+/2 (1 +k)¹⁺,

(2.16)

the last inequality being an application of (2.8). (2.15) follows.

Turning to the right-hand side of (2.12), we see that it remains to derive an upper bound on

X

x∈H⁻_`+k

e^−c²^|x−`v|

2

` −νd`+k(x)

. X

|y|>^k|v|₂

e^−c²^|y|

2

` + X

|y|≤^k|v|₂

e^−νd^k^(y). (2.17)

The first sum above is treated as in (2.16). On the other hand, the second sum is bounded above as.e^−ν⁰^k, uniformly in allksufficiently large. Sincee^−ν⁰^k .(1+k)⁻¹⁻, the bound (2.15) forE

Z_`⁽ⁿ⁾−E(Z_`⁽ⁿ⁾| A`+k)²

follows as well.

As an application of (2.15) we derive the following convergence result:

Lemma 2.6. Assume that, for someν⁰ >0, the asymptotic bound (2.15) is, uniformly innand`≤n, satisfied withˆa⁽ⁿ⁾_` .e^−ν⁰^(n−`). Then

n→∞lim X

`≤n

Z_`⁽ⁿ⁾= 0, (2.18)

P-a.s. and inL². In particular, assume that the asymptotic bound(2.10)is satisfied for an array

b⁽ⁿ⁾_x,`(y, m) with someν >0andˆb⁽ⁿ⁾_x,` .1. Then

n→∞lim X

`≤n

X

x

t^ω_x,` X

m>n−`

X

y

b⁽ⁿ⁾_x,`(y, m) f_y,m^θ^x^ω−fy,m

∆

= lim

n→∞

X

`≤n

Z_`⁽ⁿ⁾= 0, (2.19)

P-a.s. and inL²^. Proof. By (2.15),

E

E(Z_`⁽ⁿ⁾| A`−k)2 , E

Z_`⁽ⁿ⁾−E(Z_`⁽ⁿ⁾| A`+k)2

. e^−2ν⁰^(n−`)

`^d/2−(1 +k)¹⁺ ^(2.20) Applying (2.6) for eachn= 1,2, . . . (withψ²_` = ψ_`⁽ⁿ⁾2

= ^e^−2ν

0(n−`)

`^d/2− ), we infer that E X

`≤n

Z_`⁽ⁿ⁾2 .

n

X

`=1

e^−2ν⁰^(n−`)

`^d/2− .

Sinced≥3and <1/2, this implies thatP

nE P

`≤nZ_`⁽ⁿ⁾2

<∞.

Consider now the left-hand side of (2.19). For each`≤n, theZ_`⁽ⁿ⁾-sum on the right-hand side of (2.19) can be rewritten in the form (2.9) witha⁽ⁿ⁾_x,`(y, m) =b⁽ⁿ⁾_x,`(y, m)1{m>n−`}. In this case, the inequality (2.10) is satisfied for the arrayn

a⁽ⁿ⁾_x,`(y, m)o

with anyν⁰ < ν/2 andaˆ⁽ⁿ⁾_x,` .e−ν(n−`)/2 ∆= ˆa⁽ⁿ⁾_` .

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2.2 Multi-Dimensional Renewal and Asymptotics oft^ω_n Let us turn to the quenched asymptotics oft^ω_n. By construction,

t^ω_z,n=

n−1

X

m=0

X

x

t^ω_x,mf_z−x,n−m^θ^x^ω and t^ω_n=X

z

t^ω_z,n. (2.21) The claim (1.17) of Theorem B follows from:

Theorem 2.7. Assume that (2.4)holds. Then,

n→∞lim t^ω_n = 1 κ

1 +X

x,y

t^ω_x f_y−x^θ^x^ω−fy−xo_∆

= 1

κs^ω∈(0,∞), (2.22) P-a.s. and inL2on the eventn

0∈Cl^h_∞(V)o .

Proof. Part of the proof appeared in Subsection 5.3 of the review paper [12]. We rely on an expansion similar to the one employed by Sinai [16] and rewrite (2.21) as (see the beginning of Section 5.3 of [12] for details)

t^ω_z,n=tz,n+

n−1

X

`=0 n−`

X

m=1 n−`−m

X

r=0

X

x,y

t^ω_x,`

f_y−x,m^θ^x^ω −f_y−x,m

t_z−y,r. (2.23)

In this way,t^ω_n ((56) in Section 5.3 of [12]) can be represented as t^ω_n = 1

κs^ω_n+^ω_n+ t_n− 1 κ

(2.24)

where

s^ω_n= 1 +X

`≤n

X

x

t^ω_x,` f^θ^x^ω−1

, (2.25)

and the correction term^ω_n =−^ω_n,1+^ω_n,2is given by ^ω_n=−1

κ X

`≤n m>n−`

X

x

t^ω_x,` f_m^θ^x^ω−f_m

+ X

`+m+r=n

X

x

t^ω_x,` f_m^θ^x^ω−f_m t_r−1

κ

. (2.26)

By (2.1)tn−¹_κ tends to zero. We claim that,P^-a.s.,

n→∞lim s^ω_n =s^ω and X

n

E[(^ω_n)²]<∞. (2.27)

Convergence of s^ω_n. Following the discussion in Subsection 4.5 of [11], one readily verifies thats^ω>0on the eventn

0∈Cl^h_∞(V)o

. It remains to check (2.27).

Let us rewrites^ω_n as

s^ω_n−1 =X

`≤n

X

x

t^ω_x,` f^θ^x^ω−1∆

=

n

X

`=0

Z_`. (2.28)

The representation complies with (2.9) and (2.10) withˆa⁽ⁿ⁾_x,` .1 and any positiveν < c2. Hence, by (2.15),

E

E(Z_`| A_`−k)² , E

Z_`−E(Z_`| A_`+k)² . 1

`^d/2− · 1

(1 +k)¹⁺. (2.29) Sinced≥3 and <1/2, Remark 2.3 applies andlim_n→∞s^ω_n = 1 +P∞

0 Z`converges P-a.s. and inL²^.

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The^ω_n term. Let us turn now to the correction term^ω_n in (2.26). The first summand to estimate is

^ω_n,1=X

`≤n

X

x

t^ω_x,` X

m>n−`

f_m^θ^x^ω−fm

(2.30) It tends to zero by Lemma 2.6. The second summand is

^ω_n,2= X

`+m+r=n

X

x

t^ω_x,` f_m^θ^x^ω−fm

tr−1 κ

Sincetr−1/κis exponentially decaying inr, it is easy to see that (2.10) still holds with a⁽ⁿ⁾_x,` .ˆa⁽ⁿ⁾_` = e^∆ ^−c⁴^(n−`), for any positiveν < c₂and somec₄=c₄(β)>0, and Lemma 2.6 applies.

2.3 Quenched CLT

To facilitate notation setα_n=α/√

n. Forr= 1,2, . . . define S_r^ω(α)=^∆X

z

t^ω_z,re^{iα·(z−rv)}

We are studyingS_n^ω(α_n). The asymptotics ofS_n(α_n) = ES_n^ω(α_n)is given in (2.3). Us- ing (2.23),

S_n^ω(α_n) =S_n(α_n) + X

`+m+r=n

X

x,y,z

t^ω_x,` f_y−x,m^α^x^ω −f_y−x,m

t_z−y,re^{i(z−nv)·α}ⁿ. (2.31) Define

g_m^ω(α) =X

y

e^{i(y−mv)·α} f_y,m^ω −f_y,m

. (2.32)

Note thatg_m^ω(0) = f_m^ω−fm and thatg_m^ω(α)−g^ω_m(0) =P

y e^{i(y−mv)·α}−1

f_y,m^ω −fy,m . We can rewrite (2.31) as

S_n^ω(αn) =Sn(αn) + X

`+m+r=n

Sr(αn)X

x

t^ω_x,`e^{i(x−`v)·α}ⁿg_m^θ^x^ω(αn). (2.33) Expanding terms in the productsS_r(α_n)e^{i(x−`v)·α}ⁿg_m^θ^x^ω(α_n)as

S_r(α_n) =S_n(α_n) + (S_r(α_n)−S_n(α_n)) and, accordingly,

e^{i(x−`v)·α}ⁿ= 1 +

e^{i(x−`v)·α}ⁿ−1

, g_m^ω(α_n) =g^ω_m(0) + (g^ω_m(α_n)−g_m^ω(0)), we rewrite (2.33) as:

S_n^ω(αn) =Sn(αn)

1 + X

`+m≤n

X

x

+Sn(αn) X

`+m≤n

X

x

t^ω_x,` g^θ_m^x^ω(αn)−g_m^θ^x^ω(0)

+ X

`+m+r=n

(Sr(αn)−Sn(αn))X

x

+Sn(αn) X

`+m≤n

X

x

t^ω_x,`

e^{i(x−`v)·α}ⁿ−1

f_m^θ^x^ω−fm

+cross-terms

=∆Sn(αn)

1 + X

`+m≤n

X

x

+

3

X

i=1

η^ω_n,i+cross-terms.

(2.34)

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By Theorem 2.7 the sequence of random factors ofS_n(α)tend tos^ω. The cross terms are of lower order and we shall briefly discuss them at the end of the present section.

The crux of the matter is to prove:

Theorem 2.8. For everyα∈R^dthe correction termsη^ω_n,iin(2.34)satisfy : Fori= 1,2,3 lim

n→∞η^ω_n,i= 0 P-a.s. and inL²(Ω). (2.35) Once (2.35) is established, we readily infer from (2.1), (2.3) and (1.17) that

n→∞lim

S_n^ω(α/√ n)

t^ω_n = exp

−¹₂Σα·α , (2.36)

P-a.s. on the eventn

0∈Cl^h_∞(V)o

for everyα∈R^d+1 fixed. This is precisely (1.18) of Theorem B.

3 Correction Terms

In this Section, we prove (2.35). The correction termsη^ω_n,i;i= 1,2,3,will be treated separately. Recall that we are working with <1/6such that (2.4) holds withρ=/2. Theη_n,1^ω term . Consider

η^ω_n,1

S_n(α_n) =X

`≤n

X

x

t^ω_x,` X

m≤n−`

X

y

e^{i(y−mv)·α}ⁿ−1

f_y,m^θ^x^ω−fy,m

By Lemma 2.6, the constraintm ≤n−`might be removed, and we need to prove the convergence to zero of

ˆ

η^ω_n,1=^∆X

`≤n

X

x

t^ω_x,`X

m,y

a⁽ⁿ⁾_x,`(y, m) f_y,m^θ^x^ω−f_y,m∆

=X

`≤n

Z_`⁽ⁿ⁾. (3.1)

witha⁽ⁿ⁾_x,`(y, m) = e^{i(y−mv)·α}ⁿ−1 .

Lemma 3.1. In the very weak disorder regime,

n→∞lim ηˆ^ω_n,1= 0, (3.2)

P-a.s. and inL2for eachα∈R^d^fixed.

Proof of Lemma 3.1. Fora⁽ⁿ⁾_x,`(y, m)as above, (2.10) is satisfied with ˆa⁽ⁿ⁾_x,` .aˆ⁽ⁿ⁾_` = 1/^∆ √ n and anyν < c₂. By (2.15) of Lemma 2.5,

E

E(Z_`⁽ⁿ⁾| A`−k)2 , E

Z_`⁽ⁿ⁾−E(Z_`⁽ⁿ⁾| A`+k)2 . 1

n`^d/2− · 1

(1 +k)¹⁺. (3.3) By (2.6),Var ˆη_n,1^ω

.1/n. Consequently, the lacunary sequence ˆ

η^ω_n_1+δ_,1 converges to zeroP-a.s. and inL2for anyδ >0.

It remains to chooseδ > 0 appropriately and to control fluctuations of ηˆ_·,1^ω on the intervals of the form[N, . . . , N+R]with

N ∼=n^1+δ and R∼= (1 +n)^1+δ−n^1+δ ∼=n^δ. (3.4) Now,

ˆ

η^ω_N+r,1−ηˆ^ω_N,1= X

`≤N

Z_`^(N+r)−Z_`^(N) +

N+r

X

`=N+1

Z_`^(N^+r). (3.5)

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We should not worry about the second term above: (2.6) can still be applied to bound Var PN+r

`=N+1Z_`^(N^+r)

for eachrfixed. By (3.3) and the union bound,

Eh maxr≤R

N+r

X

`=N+1

Z_`^(N+r)²i

≤

R

X

r=1

Eh ^NX^+r

`=N+1

Z_`^(N+r)²i

. R N^d/2−

∼= 1

n^(1+δ)(^d2−−_1+δ^δ ). The right-hand side above is summable (inn) by our choice (3.4) whenever^d₂−−_1+δ^δ >

1. Sinced≥3and <1/2, there are feasible choices ofδ >0to ensure the latter.

As for the first term in (3.5), note that for`≤N, a^(N+r)_x,` (y, m)−a^(N_x,`⁾(y, m) =

e^{i(y−mv)·α}^N+r−e^{i(y−mv)·α}^N_∆

=b^(N,r)_x,` (y, m). (3.6) The array

b^(N,r)_x,` (y, m) satisfies (2.10) with ˆb^(N,r)_x,` .ˆb^(N,r)_` =^∆ r/N^3/2 and anyν < c2. By (2.15), (2.6) and the union bound,

Eh maxr≤R

X

`≤N

Z_`^(N^+r)−Z_`^(N⁾²i . R³

N³.

By our choice (3.4), _N^R³₃ =^∼ n⁻³for any choice ofδ >0, and, consequently, the right-hand side above is summable.

Theη_n,2^ω term. By (2.1), Sr(αn)

S_n(α_n)= tr(iαn)e^−irv·αⁿ

t_n(iα_n)e^−inv·αⁿ = e^{(r−n)(µ(iα}ⁿ^)−iv·αⁿ⁾ 1 +o(e^−c⁴^r)

. (3.7)

Setφ(α) =iv·α−µ(iα). The functionφis defined in a neighbourhood of the origin and it is of quadratic growth there. By Lemma 2.6, the residual termo (e^−c⁴^r)is negligible.

Next, for`≤nthe coefficients

a⁽ⁿ⁾_x,`(y, m) = e^(m+`)φ(αⁿ⁾−1 (3.8) satisfy (2.10) withaˆ⁽ⁿ⁾_x,` .ˆa⁽ⁿ⁾_` =^∆`/nand anyν < c2. Consequently, (2.19) enables to lift the restrictionm≤n−`. Therefore, we need to prove convergence to zero of

ˆ

η^ω_n,2=X

`≤n

X

x

t^ω_x,`X

m

e^(m+`)φ(αⁿ⁾−1

f_m^θ^x^ω−fm∆

=X

`≤n

Z_`⁽ⁿ⁾. (3.9)

Lemma 3.2. In the very weak disorder regime

n→∞lim ηˆ^ω_n,2= 0, (3.10)

P-a.s. and inL²^{for each}α∈R^d^fixed.

Proof of Lemma 3.2. Recall that for ina⁽ⁿ⁾_x,`(y, m)defined in (3.8) the asymptotic bound (2.10) is satisfied withˆa⁽ⁿ⁾_x,` .`/nuniformly inn,`≤nandx. By (2.15) and (2.6),

Var ˆη_n,2^ω . 1

n² X

`≤n

1

`^d/2−2−

=∼ 1

n^d²⁻¹⁻∧n², (3.11) which already implies the claim of Lemma 3.2 in dimensionsd≥5. We shall continue discussion for the most difficult case ofd= 3. (3.11) implies that

EX

n

ηˆ^ω_n2+δ,2

²

<∞ ⇒ lim

n→∞ηˆ_n^ω2+δ,2= 0 P−a.s.and inL2, (3.12)

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whenever

(2 +δ) ¹₂−

>1 , that is, δ > 4

1−2. (3.13)

Since < 1/6, there are choices of δ ∈ (0,1)which comply with (3.13). We need to control fluctuations ofηˆ_N^ω_+r,2−ηˆ_N,2^ω on the intervals of the form[N, . . . , N+R], where

N ∼=n^2+δ and R∼= (n+ 1)^2+δ−n^2+δ ∼=n^1+δ. (3.14) Consider the following decomposition:

ˆ

η_N^ω_+r,2−ηˆ_N,2^ω =X

`≤N

X

x

t^ω_x,`X

m

e^(m+`)φ(α^N+r⁾−e^(m+`)φ(α^N⁾

f_m^θ^x^ω−f_m +

N+r

X

`=N+1

Z_`^(N^+r)

= X

`≤N+r

X

x

t^ω_x,`X

m

e^(m+`)φ(α^N+r⁾−e^(m+`)φ(α^N⁾

f_m^θ^x^ω−fm +

N+r

X

`=N+1

Z_`^(N⁾.

(3.15) The termsZ_`^(N⁾in the second sum above were defined in (3.9) and they do not depend onr. By (2.6), we are entitled to control its maximum on the interval[N, . . . , N+R]:

Eh max

r≤R N+r

X

`=N+1

Z_`^(N)2i . 1

N²

N+R

X

`=N+1

`²

`^3/2−

∼= 1 N²

(N+R)^3/2+−N^3/2+

∼= R N^3/2−

∼= n^1+δ

n3(1+δ/2)−(2+δ) ∼= 1 n²⁺^δ²^−(2+δ)

=∆an,

(3.16)

by our choice of parameters (3.14).

For eachr ≤Rthe first term in (3.15) corresponds to the following choice of coefficients in the representation (2.9): a^(N_x,`^+r)(y, m) = e^(m+`)φ(α^N+r⁾−e^(m+`)φ(α^N⁾

. Thus, (2.10) is satisfied withˆa^(N+r)_x,` .ˆa^(N_` ^+r)^∆=`r/N²and anyν < c₂. By the very same (2.15) and (2.6), we infer that, for anyr≤R,

Var^N+rX

`=1

X

x

t^ω_x,`X

m

e^(m+`)φ(α^N+r⁾−e^(m+`)φ(α^N⁾ . r²

N⁴

N+r

X

`=1

`¹²⁺. R²

N^5/2−. (3.17) Hence, by the union bound and our choice of parameters (3.14),

Eh maxr≤R

^N+rX

`=1

X

x

t^ω_x,`X

m

e^(m+`)φ(α^N+r⁾−e^(m+`)φ(α^N⁾2i

. R³ N^5/2−

∼= 1 n²⁻^δ²^−(2+δ)

=∆bn.

(3.18)

Since <1/6, the inequality ^δ₂+ (2 +δ) <1holds for any choice ofδ≤1. Therefore, any such choice ensures thatP

n(an+bn)<∞, which implies that

n→∞lim max

n^2+δ≤r<(n+1)^2+δ

ηˆ_r,1^ω −ηˆ_n^ω2+δ,1

= 0, P-a.s and inL². The proof of Lemma 3.2 is completed.