An Almost-Sure CLT for Stretched Polymers

Article

Reference

An Almost-Sure CLT for Stretched Polymers

IOFFE, Dmitry, VELENIK, Yvan

Abstract

We prove an almost sure CLT for spatial extension of stretched polymers at very weak disorder in all dimensions d+1 larger than or equal to 4

IOFFE, Dmitry, VELENIK, Yvan. An Almost-Sure CLT for Stretched Polymers. Electronic Journal in Probability, 2013, vol. 18, no. 97, p. 1-20

arxiv : 1207.5687

DOI : 10.1214/ejp.v18-2231

Available at:

http://archive-ouverte.unige.ch/unige:21953

Disclaimer: layout of this document may differ from the published version.

arXiv:1207.5687v1 [math.PR] 24 Jul 2012

An almost sure CLT for stretched polymers

Dmitry Ioffe and Yvan Velenik Technion and Universit´e de Gen`eve

July 25, 2012

Abstract

We prove an almost sure CLT for spatial extension of stretched polymers at very weak disorder in all dimensions d+ 1≥4.

1 Introduction and Results

Directed polymers in random media were introduced in [8] as an effective model of Ising interfaces in systems with random impurities. The precise mathematical formulation appeared in the seminal paper [10], which triggered a wave of subsequent investigations. The model of directed polymers can be described as follows. Let η= (ηk)0≤k≤nbe a nearest-neighbour path onZ^dstarting at 0, and letγ= (γk)0≤k≤n

with γk= (k, ηk) be the corresponding directed path inZ^d+1. Let also{V(x)}x∈Z^d+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 path γ under the random probability measure

µ^ω_n(γ) = (Z_n;β^ω )⁻¹ exp −β

n

X

k=1

V(γk)

(2d)⁻ⁿ,

where β ≥ 0 is the inverse temperature. The behaviour of the path γ is closely related to the behaviour of the partition function Z_n;β^ω . Namely, one distinguishes between two regimes: the weak disorder regime, in which limn→∞Z_n;β^ω /E(Zn;β^ω)>

0, P-a.s., and the strong disorder regime, in which this limit is zero. It is known [3]

that there is a sharp transition between these two regimes at an inverse temperature βc which is non-trivial whend≥3. In the weak disorder regime, the pathγ behaves diffusively, in thatγnsatisfies a CLT. Diffusivity at sufficiently small values ofβ was

∗YV was partially supported by the Swiss National Science Foundation. DI was supported by the Israeli Science Foundation grant 817/09.

first established in [10]; 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 [3].

In dimensions d ≥3 the sequence Z_n;β^ω /E(Zn;β^ω) is bounded in L₂ for all suffi- ciently small values of β. In such a situation local limit versions of the CLT, which hold in probability, were established in [17, 18].

In the case of directed polymers the disorder is always strong in dimensions d = 1,2 [4, 15] and at sufficiently low temperatures. Concerning the (nondiffusive) behaviour in the strong disorder regime, we refer the reader to [5] and references therein.

In this work, we consider diffusive behaviour in dimensions d+ 1 ≥ 4 for the related models of stretched polymers. In this case, the polymer path γ can be any nearest-neighbour path on Z^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∈Z^d+1 of i.i.d. non-negative random variables. Each visit of the path to a vertex x exerts the price e^−βV(x). The stretch is introduced in one of the following two natural ways:

• The pathγ starts at 0 and ends at a hyperplane at distance nfrom 0 and has arbitrary length. This is a model of crossing random walks in random poten- tials. In dimension d+ 1 = 2, it presumably provides a better approximation to Ising interfaces in the presence of random impurities.

• The path γ has a fixed length n, 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 [11, 9, 14].

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 [22] in low dimensions d+ 1 = 2,3 or at sufficiently low temperatures.

In the case of higher dimensions d+ 1 ≥ 4, the existence of weak disorder on the level of equality between quenched and annealed free energies was established in [7, 21].

In the crossing case, a CLT in probability was established in [12] in all dimensions d+ 1 ≥4 at 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 dimensions d+ 1≥4.

1.1 Class of Models

Polymers. For the purpose of this paper, a polymer γ = (γ0, . . . , γn) is a nearest- neighbour trajectory on the integer lattice Z^d+1. Unless stressed otherwise, γ0 is always placed at the origin. The length of the polymer is |γ| =^∆ n and its spatial extension is X(γ) =^∆ γn−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∈Z^d+1

of non-degenerate non-negative i.i.d. random variables which are normalised by 0 ∈ supp(V). There is no moment assumptions on V. The case of traps, p_∞ =^∆ P(V =∞)>0, is not excluded, but then we shall assume that p_∞ is small enough.

In particular, we shall assume that P-a.s. there is an infinite connected cluster Cl_∞(V) of the set {x:V(x)<∞} in Z^d+1. In fact, we shall assume more: Given R^d+1 ∋h6= 0 and a number δ∈(√¹

d+1,1), define the positive cone Yδ^h

=∆

x∈R^d+1 : x·h≥δ|x| |h| . (1.1) By construction, the cones Yδ^h always contain at least one lattice direction±e_i, i= 1, . . . , d+ 1. We assume that it is possible to chooseδin such a fashion that, for any h, the intersection Cl^h,δ_∞(V) = Cl^∆ _∞(V)∩ Yδ^h contains (P-a.s.) an infinite connected component. For the rest of the paper, we fix such a δ ∈ (^√_d+1¹ ,1) and use the reduced notationY^h and Cl^h_∞(V) for the corresponding cones (1.1) and percolation clusters.

Weights and Path Measures. The reference measurep(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 temperature β ≥ 0 and the external pulling force h∈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^ω_h(γ) = exp{h·X(γ)−Φβ(γ)}p(γ), (1.3) where Φβ(γ)=^∆ P

xφβ ℓγ(x)

, with ℓγ(x) denoting the local time (number of visits) of γ at x, and

φ_β(ℓ)=^∆ −logEe^−βℓV. (1.4) Note that the annealed potential is positive, non-decreasing and attractive, in the sense that

0< φβ(ℓ)≤φβ(ℓ+m)≤φβ(ℓ) +φβ(m), ∀ℓ, m∈N. (1.5)

In the sequel, we shall drop the index β from the notation, and we shall drop the index h whenever 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_h^ω(γ) =X

x

e^h·xQ^ω_n(x), (1.6)

and we use Qn(x) =^∆ EQ^ω_n(x) and Qn(h) =^∆ EQ^ω_n(h) to denote their annealed coun- terparts.

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

q_h^ω(γ)

Q^ω_n(h) and Q_n,h(γ)=^{∆ 1}_{|γ|=n}

qh(γ)

Qn(h). (1.7) Very Weak Disorder The notion of very weak disorder depends on the strength

|h| of the pulling force , dimension d≥3 and the distribution ofV. It is quantified in terms of monotone non-decreasing functions ζd : [0,∞)→R₊ of sufficiently fast decay to zero lima→0ζd(a) = 0.

Definition 1. The model of stretched polymers is in the regime of very weak disorder if d≥3 and

φβ(1)< ζd(|h|). (1.8) The family{ζ_d} will be chosen appropriately small in order to guarantee validity of a certain L₂-estimate which is formulated in Lemma2.1 below.

1.2 The Result

Fix h6= 0. Then [20, 6,11]

λ=λ(β, h)= lim^∆

n→∞

1

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

v =v(h, β)=^∆ ∇λ(h), Σ= Hess[λ](h).^∆

As shall be explained later in Subsection 2.1, v 6= 0 and the matrix Σ is positive definite. Moreover, v and Σ are the limiting spatial extension and, respectively, the diffusivity matrix for the annealed model.

Theorem A. Fix h 6= 0. Then, in the regime of very weak disorder, the following holds P-a.s. on the event {0∈Cl_∞(V)}:

• The limit

nlim→∞

Q^ω_n(h)

Qn(h) (1.10)

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

• Q^ω_n-almost surely

nlim→∞

X(γ)

n =v. (1.11)

• For every α ∈R^d+1,

nlim→∞Q^ω_n

exp iα

√n(X(γ)−nv)

= exp

−1 2Σα·α

. (1.12)

Previously, Flury [7] had established that under the conditions of Theorem A (and some additional moment assumptions of the potential V)

nlim→∞

1

nlogQ^ω_n(h)

Qn(h) = 0 (1.13)

for on-axis exterior forces h. (1.13) was then extended to arbitrary directions h ∈ R^d+1 by Zygouras [21]. In [7], the analysis was carried out directly in the canonical ensemble of polymers with fixed lengthn. In [21], the author derives results for the conjugate ensemble of the so-called crossing random walks.

Large deviations under bothQ_n andQ^ω_n were investigated in [20,6]. 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, [20,6] do not imply a 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 settled in [11]. Therefore, (1.11) is a direct consequence of (1.13) and of the analysis of annealed canonical measures in [11].

The main new results of this work are (1.10) and (1.12). A version of TheoremA for the ensemble of crossing random walks appears in [12]. 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 [12] was only established in probability and not P-a.s. Thus, although the techniques developed in [12] are useful here, they certainly do not imply the claims of TheoremA, and an alternative approach was required.

1.3 Irreducible Decomposition and Basic Partition Func- tions

A polymer γ = (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 byT(x) the collection of all cone-confined paths leading from 0 to x, and by F(x) ⊂ T(x) the set of irreducible cone-confined paths. The basic partition functions are defined by

t^ω_x,n = e^{∆ −λn} X

γ∈T(x)

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

γ∈A(x)

1{|γ|=n}q_h^ω(γ). (1.15) We also set, accordingly,t^ω_n =^∆ P

xt^ω_x,n andf_n^ω =^∆ P

xf_x,n^ω . The annealed counterparts of all these quantities are denoted by t_x,n =^∆ Et^ω_x,n, f_x,n =^∆ Ef_x,n^ω , t_n =^∆ Et^ω_n and f_n =^∆ Ef_n^ω. As shown in [11], the collection {f_x,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)

where ν =ν(β, h)→ ∞ asβ becomes large, and infβ≥0ν(β, h)>0, for allh6= 0.

As in [12, 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. Fix h 6= 0. Then, in the regime of very weak disorder, the following holds P-a.s. on the event

0∈Cl^h_∞(V) :

• The limit

s^ω = lim^∆

n→∞

t^ω_n

t_n (1.17)

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

• For every α ∈R^d+1,

nlim→∞

1 t^ω_n

X

x

exp iα

√n ·(x−nv)

t^ω_x,n = exp

−1 2Σα·α

. (1.18)

For the rest of the paper, we shall focus on the proof of TheoremB

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.

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 fixing h6= 0 and then choosingβ > 0 small enough). Below, we shall summarize the required results from [11, 13]. The second is an L₂-type estimate on overlaps which holds for all β sufficiently 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 mixingales, 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 oft_nin the ballistic regime are not related to the strength of disorder and hold for all values of β ≥ 0 and appropriately large drifts h . In particular, for each h 6= 0 fixed, the annealed model is ballistic for all sufficiently small β. We refer to [11, 13] for the proof of the following: Fix h 6= 0; then, for all β > 0 small enough, λ(h) >0,

∇λ(h) 6= 0 and Hess[λ](h) is positive definite. Furthermore, there exist a small complex neighbourhoodU ⊂C^d+1 of the origin, an analytic functionµ(withµ(0) = 0) on U and a non-vanishing analytic function κ6= 0 on U such that:

nlim→∞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 [11] thatλ(h+z) =λ(h) +µ(z) for real z, and thus v =∇λ(h) =∇µ(0) and Σ = Hess[λ](h) = Hess[µ](0).

The annealed model satisfies a local LD upper bound: There existsc=c(β, h)>

0 such that, for all x∈ Y^h,

t_x,n =t_n(X(γ) =x)≤ 1 c√

n^d+1 exp

−c|x−nv|² n

. (2.2)

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

S_n α

√n ∆

=X

x

t_x,nexp

i α

√n ·(x−nv)

= 1

κ(0)exp

−Σα·α 2

1 +O(n^−1/2) , (2.3) with the second asymptotic equality holding uniformly in α on compact subsets of R^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, letAbe theσ-algebra generated by{V(x)}x∈A. We shall call such σ-algebras cylindrical.

Lemma 2.1. In the regime of very weak disorder there existsρ <1/12and constants c1, c2 <∞ such that the random weights (1.15) satisfy:

Et^ω_x,ℓt^ω_x′,ℓE f_y,m^θxω−f_y,m A

E

f_y^θ′^x′,m^ω′−f_y′,m^′

A

≤ c1e^{−c2(m+m′)} ℓ^d+1−ρ exp

(

−c2 |x−x^′|+|x−ℓv|²

ℓ +|x^′−ℓv|² ℓ

!) ,

(2.4)

uniformly in x, x^′, m, m^′, y, y^′, ℓand in all cylindricalσ-algebrasAsuch that botht^ω_x,ℓ and t^ω_x′,ℓ areA-measurable.

Remark 1. There is nothing sacred about the condition ρ < 1/12. We just need ρ to be sufficiently small. In fact, (2.4) holds with ρ = 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 disagree- ment 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. Let X1, X2, . . . be a sequence of zero-mean, square-integrable random variables. Let also {A^k}^∞_−∞ be a filtration of σ-algebras.

Suppose that we have chosen ǫ >0 and numbers a₁, a₂, . . .in such a way that E

E Xℓ

A^ℓ−k

2

≤ a²_ℓ

(1 +k)^1+ǫ and E Xℓ−E Xℓ

A^ℓ+k2

≤ a²_ℓ (1 +k)^1+ǫ

(2.5) for all ℓ = 1,2, . . . and k ≥0. Then [16] there exists K =K(ǫ)<∞ such that, for all n1 ≤n2,

E max

n1≤r≤n2

r

X

n1

Xℓ

!2

≤K

n2

X

n1

a²_ℓ. (2.6)

In particular, if P

ℓa²_ℓ <∞, then P

ℓXℓ converges P-a.s. and in L₂.

In the sequel, we shall always work with the following filtration{A^m}. Recall that we are discussing on-axis drifts h. By lattice symmetries, the mean displacement is v = ∇λ(h) = ve1 [11, 13]. At this stage, define the hyperplanes H⁻m and the corresponding σ-algebras A^m as

H⁻m =

x∈Z^d : x·e₁ ≤mv and A^m =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 indices A and two positive sequences {aα, bα}α∈A, we say that aα .bα if there exists a constant c >0 such that aα ≤cbα for allα∈ A. We shall use aα ∼=bα if bothaα .bα and aα &bα

hold. For instance, for any ǫ >0 fixed, e^−c3k2/ℓ

ℓ^(1+ǫ)/2 . 1

(1 +k)^1+ǫ, (2.8)

where the index set A is the set of pairs of integers (k, ℓ) with k ≥0 andℓ >0.

Structure of upper bounds. Our upper bounds are based on (2.8), (2.4) (ap- plied with ρ=ǫ/2) and on (2.6). Recall thatρ <1/12, and hence ǫ <1/6.

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

ℓ≤nZ_ℓ⁽ⁿ⁾. The most general form of Z_ℓ⁽ⁿ⁾ we shall consider is Z_ℓ⁽ⁿ⁾ =X

x

t^ω_x,lX

y,m

a⁽ⁿ⁾_x,ℓ(y, m) f_y,m^θxω−f_y,m

, (2.9)

where

a⁽ⁿ⁾_x,ℓ(y, m) are uniformly bounded (in all the parameters involved) arrays of real or complex numbers. Assume that there exists another family of uniformly bounded arrays

ˆ

a⁽ⁿ⁾_x,ℓ such that X

|y|≤m

e^−c2m

a⁽ⁿ⁾_x,ℓ(y, m)

.aˆ⁽ⁿ⁾_x,ℓ, (2.10) where the constant c2 is inherited from (2.4). Since

E Z_ℓ⁽ⁿ⁾

A^ℓ−k

= X

x∈H⁻ℓ−k

t^ω_x,lX

y,m

a⁽ⁿ⁾_x,ℓ(y, m)E

f_y,m^θxω−f_y,m A^ℓ−k

, (2.11)

it follows from (2.4) (applied with ρ= ^ǫ₂) and (2.10) that E

E Z_ℓ⁽ⁿ⁾

Aℓ−k

2

. 1 ℓ^d+1−ǫ/2

X

x∈H⁻l−k

e^{−c2|x−ℓv|}

2 ℓ

ˆa⁽ⁿ⁾_x,ℓ2

. (2.12)

In particular, if ˆa⁽ⁿ⁾_x,ℓ .ˆa⁽ⁿ⁾_ℓ , the bound (2.12) reduces to E

E Z_ℓ⁽ⁿ⁾

A^ℓ−k

2

. ˆ

a⁽ⁿ⁾_ℓ 2 1

ℓ^d/2−ǫ(1 +k)^1+ǫ. (2.13) Indeed, P

x∈H⁻l−ke^{−c2|x−ℓv|}

2

ℓ .ℓ^d+12 . Thus, the non-trivial part is to check (2.13) for large values of k. In the latter case, we may assume that |x−vℓ| > ^k|₂^v| for all

x∈ H⁻ℓ−k. Consequently, the sum on the right-hand side of (2.12) is bounded above by

X

x∈H⁻ℓ−k

e^{−c2|x−vℓ|2/ℓ} . Z

|y|>^k|₂^v|

e^−c2|y|2/ldy

= Z _∞

k|v| 2

r^de^−c2r2/ldr.ℓ^(d+1)/2e^−c3k2/ℓ . ℓ^d/2+1+ǫ/2 (1 +k)^1+ǫ,

(2.14)

the last inequality being an application of (2.8). (2.13) follows.

Completely similar bounds hold for E

Z_ℓ⁽ⁿ⁾−E Z_ℓ⁽ⁿ⁾

A^ℓ+k2

. Namely, note that

Z_ℓ⁽ⁿ⁾−E

Z_ℓ⁽ⁿ⁾ A^ℓ+k

= X

x∈H⁺ℓ+k

t^ω_x,ℓX

y,m

a⁽ⁿ⁾_x,ℓ(y, m) f_y,m^θxω−f_y,m

+ X

x∈H⁻ℓ+k

t^ω_x,ℓ X

z∈H⁺ℓ+k

X

m

a⁽ⁿ⁾_x,ℓ(z−x, m) f_z^θ₋^xω_x,m−E f_z^θ₋^xω_x,m

A^ℓ+k

. (2.15)

The first term in (2.15) has exactly the same structure as the right-hand side of (2.11). On the other hand, if x∈ H⁻ℓ+r for some r < k and z ∈ H⁺ℓ+k, thenf_z^θ₋^xω_x,m can be different from zero only if m ≥c4(k−r). Therefore, we conclude from (2.4) that the upper bounds of both (2.12) and (2.13) hold forE

Z_ℓ⁽ⁿ⁾−E Z_ℓ⁽ⁿ⁾

A^ℓ+k2

. Remark 2. In the sequel, we shall repeatedly derive versions of (2.12)and (2.12)for various fluctuation quantities. We shall tacitly restrict attention only to verification of the first of the conditions in (2.5)

The following easy consequence of (2.13) is useful.

Lemma 2.2. Assume that

a⁽ⁿ⁾_x,ℓ(y, m) is a bounded array. Then

nlim→∞

X

ℓ≤n

X

x

t^ω_x,l X

m>n−ℓ

X

y

a⁽ⁿ⁾_x,ℓ(y, m) f_y,m^θxω−f_y,m

= 0, (2.16)

P-a.s. and in L₂.

Proof. In this case, the inequality (2.10) is satisfied with ˆa⁽ⁿ⁾_ℓ . e^{−c2(n−ℓ)/2}. There- fore, by (2.13),

E E

Z_ℓ⁽ⁿ⁾ A^ℓ−k

2

. e^{−c2(n−ℓ)} ℓ^d/2−ǫ(1 +k)^1+ǫ Since d≥3 and ǫ <1/6, (2.6) implies thatP

nE P

ℓ≤nZ_ℓ⁽ⁿ⁾2

<∞.

2.2 Multi-Dimensional Renewal and Asymptotics of t

Let us turn to the quenched asymptotics of t^ω_n. By construction, t^ω_z,n=

n−1

X

m=0

X

x

t^ω_x,mf_z^θ₋^xω_x,n−m and t^ω_n =X

z

t^ω_z,n. (2.17) The claim (1.17) of Theorem B follows from:

Theorem 2.1. Assume that (2.4) holds. Then,

nlim→∞t^ω_n = 1

κ 1 +X

x,y

t^ω_x f_y^θ₋^xω_x−f_y−x

!

=∆ 1

κs^ω ∈(0,∞), (2.18) P-a.s. and in L₂ on the event

0∈Cl^h_∞(V) .

Proof. Part of the proof appeared in Subsection 5.3 of the review paper [13]. We rely on an expansion similar to the one employed by Sinai [17] and rewrite (2.17) as (see [13] for details)

t^ω_z,n =t_z,n+

n−1

X

ℓ=0 n−ℓ

X

m=1 n−ℓ−m

X

r=0

X

x,y

t^ω_x,l f_y^θ₋^xω_x,m−f_y−x,m

t_z−y,r. (2.19)

In this way t^ω_n (again see [13] for details) can be represented as t^ω_n = 1

κs^ω_n+ǫ^ω_n+

t_n− 1 κ

(2.20) where

s^ω_n = 1 +X

l≤n

X

x

t^ω_x,l f^θxω−1

, (2.21)

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

κ X

l≤n m>n−l

X

x

t^ω_x,l f_m^θxω−f_m

+ X

l+m+r=n

X

x

t^ω_x,l f_m^θxω−f_m

t_r− 1 κ

. (2.22)

By (2.1) t_n− κ¹ tends to zero. We claim that, P-a.s.,

nlim→∞s^ω_n =s^ω and X

n

E(ǫ^ω_n)² <∞. (2.23)

Convergence ofs^ω_n. Following the discussion in Subsection 4.5 of [12], one readily verifies that s^ω >0 on the event

0∈Cl^h_∞(V) . It remains to check (2.23).

Let us rewrites^ω_n as

s^ω_n−1 =X

ℓ≤n

X

x

t^ω_x,ℓ f^θxω−1 ∆

=

n

X

ℓ=0

X_ℓ. (2.24)

The representation complies with (2.9) and (2.10) with ˆa⁽ⁿ⁾_x,ℓ .1. Hence, by (2.13), E E Xℓ

A^ℓ−k

2

. 1

ℓ^d/2−ǫ · 1

(1 +k)^1+ǫ. (2.25)

As a result (see Remark2), (2.5) applies and, sinced≥3 and ǫ <1/6, limn→∞s^ω_n = 1 +P_∞

0 Xℓ converges P-a.s. and in L₂ .

The ǫ^ω_n term. Let us turn now to the correction term ǫ^ω_n in (2.22). The first summand to estimate is

ǫ^ω_n,1 =X

ℓ≤n

X

x

t^ω_x,ℓ X

m>n−l

f_m^θxω−f_m

(2.26) It tends to zero by Lemma 2.2. The second summand is

ǫ^ω_n,2 = X

l+m+r=n

X

x

t^ω_x,l f_m^θxω−f_m

t_r− 1 κ

Sincet_r−1/κis exponentially decaying inr, it is easy to see that there existsc3 >0 such that (2.10) still holds with ˆa⁽ⁿ⁾_ℓ .e^{−c3(n−ℓ)}, and, consequently, the computation of Lemma 2.2 still applies.

2.3 Quenched CLT

To facilitate notation set α_n=α/√n. For r = 1,2, . . . define Sr^ω(α)=^∆ X

z

t^ω_z,re^{iα·(z−rv)}

We are studying Sn^ω(α_n). The asymptotics of S_n(α_n) = ESn^ω(α_n) is given in (2.3).

Using (2.19),

Sn^ω(α_n) =S_n(α_n) + X

l+m+r=n

X

x,y,z

t^ω_x,l f_y^α₋^x_x,m^ω −f_y−x,m

t_z−y,re^{i(z−nv)·αn} (2.27) Define

g_m^ω(α) =X

y,m

e^{i(y−mv)·α} f_y,m^ω −f_y,m

. (2.28)

Note that g_m^ω(0) =f_m^ω −f_m. We can rewrite (2.27) as Sn^ω(αn) =S_n(αn) + X

l+m+r=n

S_r(αn)X

x

t^ω_x,ℓe^{i(x−ℓv)·αn}g_m^θxω(αn). (2.29) Expanding terms in the products S_r(αn)e^{i(x−ℓv)·αn}g_m^θxω(αn) as

S_r(αn) =S_n(αn) + (Sr(αn)−S_n(αn)) and, accordingly,

e^{i(x−ℓv)·αn} = 1 + e^{i(x−ℓv)·αn} −1

, g^ω_m(α_n) =g_m^ω (0) + (g_m^ω (α_n)−g^ω_m(0)), we rewrite (2.29) as:

Sn^ω(α_n) =S_n(α_n) 1 + X

l+m≤n

X

x

t^ω_x,ℓ f_m^θxω−f_m

!

+ X

l+m+r=n

(Sr(αn)−S_n(αn))X

x

t^ω_x,ℓ f_m^θxω−f_m +S_n(αn) X

l+m≤n

X

x

t^ω_x,ℓ g^θ_m^xω(αn)−g_m^θxω(0) +S_n(αn) X

l+m≤n

X

x

t^ω_x,ℓ e^{i(x−ℓv)·αn}−1

f_m^θxω−f_m

+ cross-terms

=∆ S_n(α_n) 1 + X

l+m≤n

X

x

t^ω_x,ℓ f_m^θxω−f_m

! +

3

X

i=1

η_n,i^ω + cross-terms.

(2.30)

By Theorem 2.1 random prefactors in front of S_n(α) tend to s^ω. The cross terms are of the lower order and we shall ignore them. The crux of the matter is to prove:

Theorem 2.2. For every α∈R^d the correction terms η_n,i^ω in (2.30) satisfy : For i= 1,2,3 lim

n→∞η^ω_n,i= 0 P-a.s. and in L2(Ω). (2.31) Once (2.31) is established, we readily infer from (2.1), (2.3) and (1.17) that

nlim→∞

Sn^ω(α/√ n)

t^ω_n = exp

−1

2Σα·α

, (2.32)

P-a.s. on the event

0∈Cl^h_∞(V) for every α∈R^d+1 fixed. This is precisely (1.18) of Theorem B.

3 Correction Terms

In this Section, we prove (2.31). The correction terms η^ω_n,i; i = 1,2,3, will be treated separately. Recall that we are working with ǫ < 1/6 such that (2.4) holds with ρ=ǫ/2. For the rest of this Section, let us fix a number δ >0 such that

0< δ

2 −ǫ(2 +δ)< δ

2+ǫ(2 +δ)<1. (3.1) The η_n,2^ω term . Consider

η^ω_n,2

S_n(αn) =X

ℓ≤n

X

x

t^ω_x,ℓ X

m≤n−ℓ

X

y

e^{i(y−mv)·αn} −1

f_y,m^θxω−f_y,m

By Lemma 2.2, the constraint m≤ n−ℓ might be removed, and we need to prove the convergence to zero of

ˆ

η^ω_n,2 =^∆ X

ℓ≤n

X

x

t^ω_x,ℓX

m,y

a⁽ⁿ⁾_x,ℓ(y, m) f_y,m^θxω−f_y,m ∆

=X

ℓ≤n

Z_ℓ⁽ⁿ⁾. with a⁽ⁿ⁾_x,ℓ(y, m) = e^{i(y−mv)·αn} −1

.

Lemma 3.1. In the very weak disorder regime,

nlim→∞ηˆ_n,2^ω = 0. (3.2)

P-a.s. and in L₂ for each α∈R^d fixed.

Fora⁽ⁿ⁾_x,ℓ(y, m) as above, (2.10) is satisfied with ˆa⁽ⁿ⁾_ℓ = 1/√

n. By (2.13) E

E Z_ℓ⁽ⁿ⁾

A^ℓ−k

2

. 1

nℓ^d/2−ǫ · 1

(1 +k)^1+ǫ. (3.3)

By (2.6)Var ˆη_n,2^ω

.1/n. Consequently, the lacunary sequencen ˆ η_n^ω1+δ,2

o

converges to zero P-a.s. and in L₂.

It remains to control fluctuations of ˆη^ω_·,2on 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,2−ηˆ^ω_N,2 =X

ℓ≤N

Z_ℓ^(N+r)−Z_ℓ^(N) +

N+r

X

ℓ=N+1

Z_ℓ^(N+r). (3.5) We should not worry about the second term above: (2.6) could still be applied to bound Var

PN+r

ℓ=N+1Z_ℓ^(N+r)

for each r fixed. By (3.3), and the union bound,

Emax

r≤R

N+r

X

ℓ=N+1

Z_ℓ^(N+r)

!2

≤

R

X

r=1

E

N+r

X

ℓ=N+1

Z_ℓ^(N+r)

!2

. R

N^d/2−ǫ ∼= 1

n¹⁺(^d−2¹δ−ǫ(1+δ)).

Since d ≥ 3, the right-hand side above is summable (in n) by our choice (3.4) and (3.1).

As for the first term in (3.5), note that for ℓ≤N,

a^(N_x,ℓ^+r)(y, m)−a^(N_x,ℓ⁾(y, m) = e^{i(y−mv)·αN+r} −e^{i(y−mv)·αN} ∆

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

b^(N,r)_x,ℓ (y, m)o

satisfies (2.10) with ˆb^(N,r)_ℓ =r/N^3/2 . By (2.13), (2.6) and the union bound

Emax

r≤R

X

ℓ≤N

Z_ℓ^(N+r)−Z_ℓ^(N)

!2

. R³ N³.

Again, the right-hand side above is summable (in n) by our choice (3.4).

The η_n,1^ω term. By (2.1), S_r(αn)

S_n(α_n) = e^{(r−n)(µ(iαn)−iv·αn)} 1 + o e^−c4r

. (3.7)

The restriction m≤n−ℓ could be lifted by Lemma2.2. Setφ(α) =iv·α−µ(iα).

The function φ is defined in a neighbourhood of the origin and it is of a quadratic growth there. We need to prove convergence to zero of

ˆ

η_n,1^ω =X

ℓ≤n

X

x

t^ω_x,ℓX

m

e^{(m+ℓ)φ(αn)}−1

f_m^θxω−f_m ∆

=X

ℓ≤n

Z_ℓ⁽ⁿ⁾. (3.8)

Lemma 3.2. In the very weak disorder regime

nlim→∞ηˆ_n,1^ω = 0. (3.9)

P-a.s. and in L₂ for each α∈R^d fixed.

Proof of Lemma 3.2. For ℓ . n the coefficients a⁽ⁿ⁾_x,ℓ(y, m) = e^{(m+ℓ)φ(αn)} − 1 sat- isfy (2.10) with ˆa⁽ⁿ⁾_ℓ = ℓ/n. By (2.13) and (2.6) this already implies the claim of Lemma 3.2 in dimensions d ≥5. We shall continue discussion for the most difficult case of d = 3. In this latter case we infer from (2.13) and (2.6):

Var ˆη_n,1^ω . 1

n² X

ℓ≤n

1

ℓ^3/2−2−ǫ ∼= 1

n^1/2−ǫ, (3.10)

Therefore, by the first of (3.1), EX

n

ˆ η_n^ω2+δ,1

2

<∞ ⇒ lim

n→∞ηˆ_n^ω2+δ,1 = 0 P−a.s. and in L₂ (3.11)

We need to control fluctuations of ˆη^ω_N+r,1−ηˆ_N,1^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. (3.12) Consider the following decomposition:

ˆ

η^ω_N+r,1−ηˆ_N,1^ω = X

ℓ≤N+r

X

x

t^ω_x,ℓX

m

e^{(m+ℓ)φ(αN+r)}−e^{(m+ℓ)φ(αN)} +

N+r

X

ℓ=N+1

Z_ℓ^(N). (3.13) Note that the second term above has the “mixingale” form and by (2.6) we are entitled to control its maximum on the interval [N, . . . , N +R]:

Emax

r≤R

N+r

X

ℓ=N+1

Z_ℓ^(N)

!2

. 1 N²

N+R

X

ℓ=N+1

ℓ²

ℓ^3/2−ǫ ∼= 1 N²

n(N +R)^3/2+ǫ−N^3/2+ǫo

∼= R

N^3/2−ǫ ∼= n^1+δ

n^{3(1+δ/2)−ǫ(2+δ)} ∼= 1 n^{2+δ2−ǫ(2+δ)}

=∆ an,

(3.14)

by our choice of parameters (3.12).

The first term in (3.13) corresponds to the following choice of coefficients in the representation (2.9): a^(N+r)_x,ℓ (y, m) = e^{(m+ℓ)φ(αN+r)}−e^{(m+ℓ)φ(αN)}

. Thus, (2.10) is satisfied with ˆa^(N_ℓ ^+r) =ℓr/N². By the very same (2.13) and (2.6), we infer that, for any r ≤R,

Var

N+r

X

ℓ=1

X

x

t^ω_x,ℓX

m

e^{(m+ℓ)φ(αN+r)}−e^{(m+ℓ)φ(αN)}

!

∼= R²

N^5/2−ǫ. (3.15) Hence, by the union bound and our choice of parameters (3.12),

Emax

r≤R N+r

X

ℓ=1

X

x

t^ω_x,ℓX

m

e^{(m+ℓ)φ(αN+r)}−e^{(m+ℓ)φ(αN)}

!²

. R³

N^5/2−ǫ ∼= 1 n^{2−δ2−ǫ(2+δ)}

=∆ bn.

(3.16)

Since, by both inequalities in (3.1),P

n(an+bn)<∞, it follows that, P-a.s.,

nlim→∞ max

n^2+δ≤r<(n+1)^2+δ

ηˆ^ω_r,1−ηˆ_n^ω2+δ,1

= 0, and the proof is completed.

The η_n,3^ω term . Recall that η_n,3^ω

S_αn =X

ℓ≤n

X

x

t^ω_x,ℓ e^{i(x−ℓv)·αn}−1 X

m≤n−ℓ

f_m^θxω−f_m .