Francis Comets
email: comets@math.univ-paris-diderot.fr

Quansheng Liu
email: quansheng.liu@univ-ubs.fr

Rate of convergence for polymers in a weak disorder

Keywords: Directed polymers, random environment, weak disorder, rate of convergence, regular martingale, central limit theorem for martingales, stable and mixing convergence AMS 2010 subject classifications: Primary 60K37. Secondary 60F05, 60G45, 60J80, 82D60

We consider directed polymers in random environment on the lattice Z d at small inverse temperature and dimension d ≥ 3. Then, the normalized partition function W n is a regular martingale with limit W . We prove that n (d-2)/4 (W n -W )/W n converges in distribution to a Gaussian law. Both the polynomial rate of convergence and the scaling with the martingale W n are different from those for polymers on trees.

1 Polymer models and statement of the main result

Motivation

We consider directed polymers in random environment, given by a simple random walk on the d-dimensional lattice in a space-time random potential. In a seminal paper, Derrida and Spohn [START_REF] Derrida | Polymers on disordered trees, spin glasses, and traveling waves[END_REF] perform a detailed analysis of polymers on the Cayley tree, or equivalently, the branching random walk with a fixed branching number. Later the same model has been taken up as an approximation and a toy model with explicit computations: in the physics literature, we mention the pleasant, recent and documented survey [START_REF] Halpin-Healy | A KPZ cocktail -shaken, not stirred . . . Toasting 30 years of kinetically roughened surfaces[END_REF], and also [START_REF] Dean | Extreme-value statistics of hierarchically correlated variables deviation from Gumbel statistics and anomalous persistence[END_REF] for the statistics of extremes on the hierarchical tree at zero temperature; on the mathematical side, the authors of [START_REF] Alberts | The near-critical scaling window for directed polymers on disordered trees[END_REF] study the near-critical scaling window on the tree, the analogue of the intermediate disorder regime where the rescaled lattice model on line converges to the KPZ continuum random polymer [START_REF] Alberts | The intermediate disorder regime for directed polymers in dimension 1+1[END_REF][START_REF] Caravenna | Polynomial chaos and scaling limits of disordered systems[END_REF]. Not only a source of inspiration and guidance, this model, as well as related random cascades, were also found to provide quantitative bounds on polymer models on the lattice in [START_REF] Comets | Majorizing multiplicative cascades for directed polymers in random media[END_REF][START_REF] Liu | Exponential inequalities for martingales and asymptotic properties of the free energy of directed polymers in random environment[END_REF][START_REF] Nguyen | Stochastic order and disordered systems[END_REF].

In spite of these similarities, the two models behave quite differently in many aspects. In the strong disorder phase, the free energy of the branching process is linear in the inverse temperature β though it is strictly convex for the polymer on the lattice, see Theorem 1.5 in [START_REF] Comets | The number of open paths in an oriented ρ-percolation model[END_REF] in the case of a Bernoulli environment. Also, the fluctuations are expected to be of a completely different nature in the two models. In this paper, we consider the weak disorder regime, and we show that the martingale convergence takes place at a polynomial rate, whereas it is exponential in the corresponding supercritical Galton-Watson process [START_REF] Heyde | A rate of convergence result for the super-critical Galton-Watson process[END_REF][START_REF] Heyde | Some central limit analogues for supercritical Galton-Watson processes[END_REF].

More precisely, it is shown in [START_REF] Heyde | A rate of convergence result for the super-critical Galton-Watson process[END_REF][START_REF] Heyde | Some central limit analogues for supercritical Galton-Watson processes[END_REF] that, for a Galton-Watson process (Z n ) with Z 0 = 1, m = EZ 1 > 1 and EZ 2 1 < ∞, the renormalized population size W n = Z n /m n is a regular martingale with limit W such that m n/2 (W -W n ) → aW 1/2 G in distribution [START_REF] Alberts | The near-critical scaling window for directed polymers on disordered trees[END_REF] and

m n/2 (W -W n ) W 1/2 n → aG in distribution, (2)
where a 2 = VarZ 1 m 2 -m , G is a Gaussian N (0, 1) distributed random variable independent of W . Similarly, for branching random walks, the convergence of the Biggins martingale to its limit is exponentially fast [START_REF] Iksanov | On the rate of convergence of a regular martingale related to a branching random walk[END_REF][START_REF] Iksanov | A central limit theorem and a law of the iterated logarithm for the Biggins martingale of the supercritical branching random walk[END_REF] in the regular case. Recently the same question was studied for a branching process in a random environment [START_REF] Huang | Convergence rates for a supercritical branching process in a random environment[END_REF][START_REF] Wang | Central limit theorems for a supercritical branching process in a random environment[END_REF], leading to similar conclusions.

In this paper, we consider random polymers on the lattice in a time-space dependent random medium, deep inside the weak disorder regime. Similar to the supercritical case of a branching process, weak disorder can be defined as the regime where the natural martingale is regular [START_REF] Bolthausen | A note on diffusion of directed polymers in a random environment[END_REF][START_REF] Imbrie | Diffusion of directed polymer in a random environment[END_REF], or where the polymer is diffusive [START_REF] Comets | Directed polymers in random environment are diffusive at weak disorder[END_REF]. It holds in space dimension d ≥ 3 [START_REF] Lacoin | New bounds for the free energy of directed polymers in dimension 1+1 and 1+2[END_REF] and at a temperature larger than some critical value which can be estimated by second moment and entropy considerations [START_REF] Birkner | A condition for weak disorder for directed polymers in random environment[END_REF][START_REF] Camanes | The critical temperature of a directed polymer in a random environmentMarkov Process[END_REF][START_REF] Hu | A note on directed polymers in Gaussian environments[END_REF]. In Theorem 1.1 below, we prove that, at large temperature, the speed of convergence is polynomial but not exponential, and the limit scales with W or W n instead of their square root as in ( 1) and [START_REF] Alberts | The intermediate disorder regime for directed polymers in dimension 1+1[END_REF]. Precisely, we show a central limit theorem for the difference between the martingale and its limit: the ratio of the difference divided by n -(d-2)/4 times the martingale is asymptotically normal.

In view of (1) and (2), this limit has two remarkable and unexpected features. The slowdown in the rate of convergence (compared to the branching case) is due to space correlations coming from further intersections between paths on the lattice but not on the tree. Also the unusual linear scaling in the martingale can be understood as coming from fluctuations, and quadratic variations scale like the square of the martingale.

Notations

• The random walk: ({S n } n≥0 , P x ) is a nearest neighbor, symmetric simple random walk on the d-dimensional integer lattice Z d starting from x, d ≥ 3. We let P = P 0 and we denote by P [f ] = f dP the expectation of f with respect to P .

• The random environment: η = {η(n, x) : n ∈ N, x ∈ Z d } is an independent and identically distributed (i.i.d.) sequence of real random variables (r.v.'s), non-constant, such that,

λ(β) := ln E[exp(βη(n, x))] < ∞ for all β ∈ R,
where we denote by E the expectation over the environment. The corresponding probability measure will be denoted by P.

• The partition function at inverse temperature β ∈ R

W n := P [exp (βH n (S) -nλ(β))] with H n (S) = 0≤t≤n-1 η(t, S t ), n ≥ 1, (3)
is the normalization which makes the Gibbs measure W n -1 exp{βH n (S)nλ(β)} dP a probability measure on the path space for a fixed realization of the environment. Note that we use a slightly different definition than usual, including t = 0 but not t = n in the Hamiltonian H n . This makes no fundamental difference in the results (see Remark 1.2 below), but it yields simpler formulas here: for two independent simple random walks S = (S t ) and S = ( S t ), we have Cov P (H n (S), H n ( S)) = Var(η(0, 0)) N n , where Cov P denotes the covariance with respect to P, N n is the number of intersections of the paths S, S up to time n:

N n = N n (S, S) := n-1 t=0 1 St= St , (4)
whose limit

N ∞ = N ∞ (S, S) := ∞ t=0 1 St= St (5)
has expectation given by the standard Green function [START_REF] Iksanov | On the rate of convergence of a regular martingale related to a branching random walk[END_REF]. The sequence (W n ) depends on the environment, and it is a positive martingale with respect to the filtration

G n = σ{η(t, x); t ≤ n -1, x ∈ Z d }, n ≥ 1. ( 6
)
It is well known [START_REF] Bolthausen | A note on diffusion of directed polymers in a random environment[END_REF][START_REF] Imbrie | Diffusion of directed polymer in a random environment[END_REF] that

W = lim n→∞ W n exists a.s., with P(W > 0) = 0 or 1.
Moreover, the convergence holds in L 2 for β in a neighborhood of 0, defined by (L2)

λ 2 := λ(2β) -2λ(β) < ln(1/π d ), (7)
where π d is the return probability of the simple random walk,

π d := P {S n = 0 for some n ≥ 1} ∈ (0, 1) (8)
by transience since d ≥ 3. Now, we give a short account of the main steps of the computation of [START_REF] Bolthausen | A note on diffusion of directed polymers in a random environment[END_REF], which is useful for the sequel. We can express W 2 n as a sum W 2 n = P ⊗2 e β[Hn(S)+Hn( S)]-2nλ(β) over independent paths (so-called replicas), and we compute, using Fubini's theorem and independence,

E[W 2 n ] = P ⊗2 n-1 t=0 E e β[η(t,St)+η(t, St)]-2λ(β) = P ⊗2 n-1 t=0 e λ(2β)-2λ(β) 1 St= St + 1 St = St = P ⊗2 n-1 t=0 e λ 2 1 S t = St = P ⊗2 e λ 2 Nn , (9)
with N n as in [START_REF] Birkner | A condition for weak disorder for directed polymers in random environment[END_REF]. As n → ∞, N n ր N ∞ = ∞ t=0 1 St= St . Since the process (S t -S) under P ⊗2 has the same law as (S 2t ) under P and S 2t+1 = 0 P -a.s., N n has the same law as the number of visits to 0 of a simple random walk in time 2n, and N ∞ is geometrically distributed with "failure" probability π d :

P ⊗2 (N ∞ = k) = π k-1 d (1 -π d ) for k ≥ 1. ( 10
)
Therefore, by the monotone convergence theorem, as n → ∞,

E[W 2 n ] ր P ⊗2 e λ 2 N∞ =    (1 -π d )e λ 2 1 -π d e λ 2 if π d e λ 2 < 1, +∞ if π d e λ 2 ≥ 1.
In this paper, we always assume [START_REF] Caravenna | Polynomial chaos and scaling limits of disordered systems[END_REF], so that the martingale (W n ) is bounded in L 2 . By Doob's convergence theorem, W n → W in L 2 and then W > 0. In particular,

EW 2 = (1 -π d )e λ 2 1 -π d e λ 2 and Var (W ) = e λ 2 -1 1 -π d e λ 2 . ( 11
)
1.3 A Gaussian limit and the rate of convergence Before coming to our main result, we recall two convergence modes. Let (Y n ) be a sequence of real random variables defined on a common probability space (Ω, F , P ), converging in distribution to a limit Y .

• This convergence is called stable if for all B ∈ F with P (B) > 0, the conditional law of Y n given B converges to some probability distribution depending on B.

• This convergence is called mixing if it is stable and the limit of conditional laws does not depend on B -and therefore is the law of Y -.

The stable convergence allows to add extra variables: for any fixed r.v. Z on (Ω, F , P ), the couple (Y n , Z) converges in law to some coupling of Y and Z on an extended space. The mixing convergence means that Y n is asymptotically independent of all event A ∈ F . These convergences were introduced by Rényi [START_REF] Rényi | On stable sequences of events[END_REF]; we refer to [START_REF] Aldous | On mixing and stability of limit theorems[END_REF] for a nice presentation with the main consequences, and to [START_REF] Hall | Martingale limit theory and its application[END_REF] pp. 56-57 for an extended account on the connections with martingale central limit theorem.

Theorem 1.1 For d ≥ 3, there exists some β 0 > 0 such that, for |β| < β 0 ,

n d-2 4 (W -W n ) → σ 1 W G in distribution (12)
and

n d-2 4 (W -W n ) W n → σ 1 G in distribution, (13)
where σ 1 is from [START_REF] Vargas | A Local limit theorem for directed polymers in random media: the continuous and the discrete case[END_REF], G is a Gaussian r.v. with law N(0, 1), which is independent of W . Moreover, the convergence in (12) is stable, and the convergence in (13) is mixing.

The theorem calls for some comments. The value of β 0 is defined by the conditions in Lemma 4.1, Lemma 2.3 (b) and Lemma 3.4. The result is quite different from (1)- [START_REF] Alberts | The intermediate disorder regime for directed polymers in dimension 1+1[END_REF]. The speed of convergence of W n to its limit is n -d-2 4 . The mixing convergence in [START_REF] Giacomin | Random polymer models[END_REF] shows that the random variable G n defined in the left hand side of (13) is asymptotically independent of each event A of the environmental probability space, in the sense that for all real y, we have lim n→∞ P ({G n ≤ y} ∩ A) = P (G ≤ y)P (A). Our approach relies on a central limit theorem for martingales. Remark 1.2 (Usual Hamiltonian) For the standard Hamiltonian collecting the environment at times 1, 2, . . . n,

W n := P exp β 1≤t≤n η(t, S t ) -nλ(β) = W n+1 exp{-βη(0, 0) + λ(β)} , it is straightforward to see that, for d ≥ 3 and |β| < β 0 , W n → W := W exp{-βη(0, 0)+λ(β)}, that n d-2 4 (W -W n ) → σ 1 W G in distribution and n d-2 4 (W -W n ) W n → σ 1 G in distribution,
with σ 1 and G as above and G independent from W .

Organization of the paper: In Section 2 we start with algebraic computations of covariances and use Green function estimates to derive asymptotics. Of independent interest, a specific form of central limit theorem for infinite martingales arrays is given in Lemma 3.1 of Section 3, and used to prove Theorem 1.1. The proofs of intermediate lemmas are postponed to Section 4. A key step consists in controling the sum of conditional variances in the quadratic norm, so we implement the necessary algebra for a system of 4 replicas at the beginning of the section.

The correlation structure

It is useful to introduce

W n (x) = P e βHn(S)-nλ(β) 1 Sn=x ,

and to observe that, for m ≥ 0 including m = ∞ with the convention W ∞ = W ,

W n+m = x W n (x)W m • θ n,x , (15)
by Markov property. We view W n as a function of η, we denote by θ n,x the shift operator on the environment, θ n,x η : (t, y) → η(n + t, x + y). We use θ x = θ 0,x as a short notation. Taking m = ∞ we see that

W -W n = x W n (x) W • θ n,x -1 . (16)
For nearest-neighbor paths S, S define τ n the time delay of first intersection after time n ≥ 0,

τ n (S, S) = inf{k ≥ 0 : S n+k = S n+k },
with the convention that inf ∅ = +∞. We write τ = τ 0 , and we note that π d = P ⊗2 0,0 (τ 1 < ∞).

Covariance of the martingale limit and rate of convergence in L 2

Recall that Var

(W ) = e λ 2 -1 1 -P ⊗2 0,0 (τ 1 < ∞)e λ 2 , cf. (11). Proposition 2.1 Assume λ 2 < 2 d ln 1 π d . Then, Cov(W, W • θ x ) = Var(W ) × P ⊗2 0,x (τ < ∞) . (17)
Moreover,

W -W n 2 2 = Var(W ) × P ⊗2 0,0 (e λ 2 Nn 1 τn<∞ ), (18)
and, as n → ∞ ,

W -W n 2 2 ∼ σ 2 × n -(d-2)/2 , ( 19
)
with the constant σ 2 from (28).

Proof. We first compute the covariance of W and W • θ x . Denote by F n the σ-field generated by S i , S i for 0 ≤ i ≤ n. By convergence in L 2 , we obtain as in [START_REF] Comets | Majorizing multiplicative cascades for directed polymers in random media[END_REF],

Cov(W, W • θ x ) = lim m→∞ E (W m -1)(W m • θ x -1) = lim m→∞ P ⊗2 0,x E (e βHm(S)-mλ(β) -1)(e βHm( S)-mλ(β) -1) (Fubini) = lim m→∞ P ⊗2 0,x E e βHm(S)+βHm( S)-2mλ(β) -1 = lim m→∞ P ⊗2 0,x e λ 2 Nm -1 = P ⊗2 0,x (e λ 2 N∞ -1) = P ⊗2 0,x 1 τ <∞ P ⊗2 0,x (e λ 2 N∞ -1)|F τ = P ⊗2 0,x (τ < ∞)P ⊗2 0,0 (e λ 2 N∞ -1) (strong Markov property)
which is the first claim since Var(W ) = P ⊗2 0,0 e λ 2 N∞ -1. We next calculate the L 2 norm of W -W n . By [START_REF] Heyde | A rate of convergence result for the super-critical Galton-Watson process[END_REF],

W -W n 2 2 = E x W n (x)(W • θ n,x -1) 2 = E x,y W n (x)W n (y)(W • θ n,x -1)(W • θ n,y -1) = x,y E[W n (x)W n (y)]E[(W • θ n,x -1)(W • θ n,y -1)] (independence) = Var(W ) x,y E [W n (x)W n (y)] P ⊗2 x,y (τ < ∞) (by (17))
= Var(W )

x,y

P ⊗2 0,0 e λ 2 Nn 1 {Sn=x, Sn=y} P ⊗2 x,y (τ < ∞) (cf. (9)) = Var(W ) × P ⊗2 0,0 (e λ 2 Nn 1 τn<∞ ),
by Markov property. This is [START_REF] Hu | A note on directed polymers in Gaussian environments[END_REF]. We finally derive (19) using classical estimates for the Green function. Denote by

G(x) := P ⊗2 0,x ∞ n=0 1 {Sn-Sn=0} = P ⊗2 0,x (N ∞ ) (20)
the Green function for the symmetrized walk (S n -S n ) n , and observe that it is equal on even sites x (i.e., when x 1 = 0 mod 2) to the Green function of the simple random walk: for even sites x

G(x) = P x ∞ n=0 1 Sn=0 ,
since the process (S n -S n ) under P ⊗2 0,x has the same law as (S 2n ) under P x and S 2n+1 = 0 P x a.s. (for even sites x). For odd sites x, G(x) = 0, since in this case S n -S n = 0 P ⊗2 0,x a.s.. From the geometric distribution of N ∞ under P ⊗2 0,0 , we have

G(0) = (1 -π d ) -1
. By Markov property we have

P ⊗2 0,x (τ < ∞) = G(x)/G(0), x ∈ Z d ,
and can use the classical estimates for the Green function, see e.g. [25, Theorem 4.3.1]: for even sites x,

G(x) = K d |x| d-2 + O 1 |x| d , |x| → ∞, (21)
where

K d ∈ (0, ∞) is a constant whose value is K d = d Γ(d/2) (d -2)π d/2 . ( 22
)
By the central limit theorem, we have the following convergence in distribution under P ⊗2 0,0 to a Gaussian vector:

n -1/2 (S n -S n ) → Z in distribution,
where Z is a Gaussian vector with mean 0 and covariance matrix (2/d)I d . Thus, with [START_REF] Iksanov | A central limit theorem and a law of the iterated logarithm for the Biggins martingale of the supercritical branching random walk[END_REF],

n (d-2)/2 G(S n -S n ) → K d |Z| d-2 in distribution. ( 23
)
We shall need the following two lemmas, whose proofs are postponed by the end of the section.

Lemma 2.2 (Asymptotic independence) Under P ⊗2 we have the following convergence in distribution of random vectors:

N n , n -1/2 S n , n -1/2 S n -→ (N, Z 1 , Z 1 ) in distribution,
where N, Z 1 , Z 1 are independent, and where

• N is geometrically distributed with parameter 1-π d : P (N = k) = π k-1 d (1-π d ) for k ≥ 1,
• Z 1 and Z 1 are Gaussian vectors with mean 0 and covariance matrix

(1/d)I d . Lemma 2.3 (Boundedness in L 1+δ ) (a) For a > 0, lim sup n→∞ P ⊗2 S n -Sn √ n -a 1 {Sn-Sn =0} < ∞ if and only if a < d. ( 24
) (b) If λ 2 < 2 d ln 1 π d , then for δ > 0 small enough, lim sup n→∞ P ⊗2 [e λ 2 Nn n d-2 2 G(S n -Sn )] 1+δ < ∞. ( 25
)
We first end the proof of Proposition 2.1. From Lemma 2.2, with

Z = Z 1 -Z 1 , e λ 2 Nn × n d-2 2 G(S n -S n ) law -→ e λ 2 N × K d |Z| d-2 . ( 26
)
By Lemma 2.3(b), the sequence in the left-hand side of ( 26) is uniformly integrable, so that the convergence in law implies the convergence of expectations, allowing to write

n d-2 2 P ⊗2 0,0 (e λ 2 Nn 1 τn<∞ ) = n d-2 2 P ⊗2 0,0 e λ 2 Nn P ⊗2 0,0 (τ n < ∞|F n ) = G(0) -1 P ⊗2 0,0 e λ 2 Nn × n d-2 2 G(S n -S n ) -→ G(0) -1 P ⊗2 0,0 e λ 2 N∞ × E K d |Z| d-2 . ( 27
)
Together with [START_REF] Hu | A note on directed polymers in Gaussian environments[END_REF], this ends the proof of ( 19), yielding the value

σ 2 = K d Z d G(0) Var(W )E(W 2 ) = K d Z d (1 -π d ) 2 × (e λ 2 -1)e λ 2 × 1 (1 -π d e λ 2 ) 2 , ( 28
)
with Var(W ) from [START_REF] Dean | Extreme-value statistics of hierarchically correlated variables deviation from Gumbel statistics and anomalous persistence[END_REF] and

Z d def = E 1 |Z| d-2 = 1 Γ(d/2) d 4 (d-2)/2 (29)
from the chi-square distribution. This proof of Proposition 2.1 is complete.

For later purposes, define

σ 2 1 = K d Z d (1 -π d ) × Var(W ), ( 30
)
so that σ 2 = σ 2 1 EW 2 .

Proof of the lemmas

It remains to prove Lemmas 2.2 and 2.3. Proof of Lemma 2.2. First observe that

sup n≥m P ⊗2 (N n > N m ) → 0 as m → ∞, since N n ր N ∞ < ∞ a.
s. Fix m ≥ 1 and f, g, g continuous and bounded. For all n ≥ m, we write

P ⊗2 f (N n )g n -1/2 S n g n -1/2 Sn = P ⊗2 f (N n )g n -1/2 S n g n -1/2 Sn 1 Nn=Nm + ǫ(n, m) = P ⊗2 f (N m )g n -1/2 S n g n -1/2 Sn 1 Nn=Nm + ǫ(n, m) = P ⊗2 f (N m )g n -1/2 (S n -S m ) g n -1/2 ( Sn -Sm ) 1 Nn=Nm + ǫ ′ (n, m) = P ⊗2 f (N m )g n -1/2 (S n -S m ) g n -1/2 ( Sn -Sm ) + ǫ"(n, m) = P ⊗2 [f (N m )] × P g n -1/2 (S n -S m ) × P g n -1/2 ( Sn -Sm ) + ǫ"(n, m) ,
where the equalities define the terms ǫ(n, m), ǫ ′ (n, m), ǫ ′′ (n, m) on their first occurrence. Here,

|ǫ(n, m)| ≤ f ∞ g ∞ g ∞ P (N n = N m )
tends to 0 as m → ∞ uniformly in n ≥ m, ǫ ′ (n, m)ǫ(n, m) → 0 as n → ∞ for all fixed m, and sup n≥m ǫ ′′ (n, m) → 0 as m → ∞. The last equality comes from independence in the increments of the random walks, and of the two random walks S and S. Hence, letting n → ∞ and then m → ∞, we get

P ⊗2 f (N n )g n -1/2 S n g n -1/2 Sn → P ⊗2 [f (N ∞ )] × P [g(Z 1 )] × P [g( Z 1 )].
Since N ∞ is geometrically distributed with parameter 1π d (see [START_REF] Comets | Directed polymers in random environment are diffusive at weak disorder[END_REF]), this ends the proof of the lemma.

Remark 2.4

We could have taken another route to prove the lemma. The couple n -1/2 (S n , S n ) converges (mixing) to the Gaussian vector (Z 1 , Z1 ), see e.g. Theorem 2 in [START_REF] Aldous | On mixing and stability of limit theorems[END_REF]. On the other hand N n → N ∞ a.s. From the mixing consequence that we mentioned above Theorem 1.1 (which remains valid for random variables with values in R d ) it follows the convergence in distribution of n -1/2 S n , n -1/2 S n , N ∞ to (Z 1 , Z1 , N). It is not difficult to see that the sequence (n -1/2 S n , n -1/2 S n , N ν ) has the same limit in distribution. However, we have given the above proof, which is short and instructive, for the sake of completeness.

Proof of Lemma 2.3. (a) The "only if" part is evident by Fatou's lemma since

S n -Sn √ n -a 1 {Sn-Sn =0} → |Z| -a in distribution
and E|Z| -a < ∞ if and only if a < d. Let's show the "if" part. Since the L a -norm is increasing in a, we only need to prove the finiteness of the lim sup for 0 < a < d sufficiently close to d. So we fix a ∈ (d -1, d). By the local central limit theorem (see e.g. [24, Theorem 1.2.1, p.14]) we know that

p n (x) := P ⊗2 (S n -Sn = x) = P (S 2n = x) satisfies |p n (x) -pn (x)| ≤ c 2 n -d/2 |x| -2 , with pn (x) = c 1 n -d/2 exp{- d|x| 2 4n }, (31)
where c 1 , c 2 > 0 are constants. (In fact we have c 1 = C d , with C d defined by (65).) Denote by I n the integral in [START_REF] Lawler | Intersections of Random Walks[END_REF]. Then

I n ≤ x =0 |x| √ n -a pn (x) + x =0 |x| √ n -a c 2 n -d/2 |x| -2 := I n,1 + I n,2 . (32)
In the following to avoid sums over non-integer valued numbers, we shall use the integer valued For the first sum in (32), we have, for some constant c 3 > 0,

L 1 norm x 1 = |x 1 | + • • • + |x d | for x = (x 1 , • • • , x d ) ∈ Z d ,
I n,1 ≤ c 1 d a n -d/2 x =0 x 1 √ n -a exp{- x 2 1 4dn } = c 1 d a n -d/2 r≥1 x 1 =r r √ n -a exp{- r 2 4dn } ≤ c 1 c 3 d a n -d/2 r≥1 r d-1 r √ n -a exp{- r 2 4dn } = c 1 c 3 d a √ n r≥1 r √ n -(a-d+1) exp{- 1 4d ( r √ n ) 2 },
where the next to last step holds as the number of x = (x 1 , ...,

x d ) ∈ Z d with |x 1 |+• • •+|x d | = r is bounded by 2(2r+1) d-1 ≤ c 3 r d-1
(notice that each coordinate satisfies |x i | ≤ r, and while the first d -1 are chosen, the absolute value of the last coordinate is determined by the equation, so that the last coordinate has at most 2 possibilities). As 0 < ad + 1 < 1, we have, for all n ≥ 1,

I n,1 ≤ c 1 c 3 d a √ n 1≤r≤ √ n ( r √ n ) -(a-d+1) + r> √ n exp{- 1 4d ( r √ n ) 2 } ≤ c 1 c 3 d a 1≤r≤ √ n r √ n r-1 √ n x -(a-d+1) dx + r> √ n r √ n r-1 √ n exp{- x 2 4d }dx ≤ c 4 := c 1 c 3 d a 1 0 x -(a-d+1) dx + ∞ 0 exp{- x 2 4d }dx < ∞.
Similarly, for the second sum in (32), we have,

I n,2 ≤ c 2 c 3 d a+2 n -d/2 r≥1 r √ n -a r -2 r d-1 = c 2 c 3 d a+2 n (a-d)/2 r≥1 r -(a+3-d)
which tends to 0 as n → ∞ (since d-1 <a < d). This ends the proof of Part (a) of Lemma 2.3.

(b) Let G n = G n (S n -Sn ) := n d-2 2 G(S n -Sn ) and λ ′ 2 = (1 + δ)λ 2 . Recall from (21) that G(x) ≤ c 5 |x| 2-d . Then P ⊗2 (e λ 2 Nn G n ) 1+δ = z P ⊗2 e λ ′ 2 Nn ; S n -S n = z G n (z) 1+δ ≤ c 5 z =0 P ⊗2 e λ ′ 2 Nn ; S n -S n = z n -1/2 z -(d-2)(1+δ) + c 5 P ⊗2 e λ ′ 2 Nn ; S n -S n = 0 n (d-2)(1+δ)/2 . ( 33
)
Denoting by L n = sup{j = 0, . . . n : S j -S j = 0} and T = inf{j ≥ 1 : S j -S j = 0} the last (before n) and the first hitting times of 0, we will use that, under Condition (L2),

P ⊗2 e λ ′ 2 Nn ; S n -S n = 0 ∼ e λ ′ 2 [1 -e λ ′ 2 π d ] -2 P ⊗2 [T = n] as n → ∞, (34)
see Theorem 2.2 (case 2) in [START_REF] Giacomin | Random polymer models[END_REF]. Since

P ⊗2 [T = n] ≤ P ⊗2 (S n -S n = 0) = O(n -d/2
), this implies that when (d -2)(1 + δ) < d, the last term in (33) vanishes as n → ∞, and yields, for z = 0,

P ⊗2 e λ ′ 2 Nn ; S n -S n = z = n j=0 P ⊗2 e λ ′ 2 Nn ; S n -S n = z, L n = j Markov = n j=0 P ⊗2 e λ ′ 2 N j ; S j -S j = 0 P ⊗2 S n-j -S n-j = z, L n-j = 0 (34) ≤ c 6 n j=0 P ⊗2 [T = j]P ⊗2 S n-j -S n-j = z, L n-j = 0 ≤ c 6 n j=0 P ⊗2 [T = j]P ⊗2 S n-j -S n-j = z ≤ c 6 P ⊗2 S n -S n = z .
Inserting this in (33), we obtain for large n,

P ⊗2 (e λ 2 Nn G n ) 1+δ = c 5 c 6 1 + z =0 P ⊗2 S n -S n = z n -1/2 z -(d-2)(1+δ)
, which, according to [START_REF] Lawler | Intersections of Random Walks[END_REF], is bounded provided that (d -2)(1 + δ) < d.

Proof of the Central Limit Theorem

In this section we give the proof of Theorem 1.1. Our proof is based on the following central limit theorem for infinite martingale arrays, which is a slight extension of Corollaries 3.1 and 3.2 of the book by Hall and Heyde [START_REF] Hall | Martingale limit theory and its application[END_REF] (pp. 58-59 and p.64), but we could not find it in the literature.

Lemma 3.1 For n ≥ 1, let {(S n,i , F n,i ) : i ≥ 0} be a martingale defined on a probability space (Ω, F , P ), with S n,0 = 0 and sup

n,i≥1 ES 2 n,i < ∞. ( 35
)
Let X n,i = S n,i -S n,i-1 , i ≥ 1 be the martingale differences, and S n,∞ = lim i→∞ S n,i be the a.s. limit of (S n,i , i ≥ 0). Suppose that:

(a) the conditional variance converges in probability: for a real random variable V ∈ [0, ∞),

V 2 n,∞ := ∞ i=1 E(X 2 n,i |F n,i-1 ) -→ V 2 in probability; (36)
(b) the conditional Lindeberg condition holds :

∀ε > 0, ∞ i=1 E(X 2 n,i 1 |X n,i |>ε |F n,i-1 ) -→ 0 in probability; ( 37
) (c) the σ -f ields are nested: F n,i ⊂ F n+1,i for all n, i ≥ 1. Then S n,∞ -→ V G in distribution ( 38
)
where G is a Gaussian variable with law N(0, 1) and independent of V ; if additionally V = 0 a.s., then

S n,∞ V n,∞ -→ G in distribution. ( 39
)
Moreover, the convergence in (38) is stable, and the convergence in (39) is mixing.

Lemma 3.1 reduces to Corollaries 3.1 and 3.2 in [START_REF] Hall | Martingale limit theory and its application[END_REF] for a triangular array of martingales differences, that is, when X n,i = 0 for all i > k n , for some sequence of integers k n increasing to ∞. As in the case of a triangular array, if V is measurable with respect to each F n,i for n, i ≥ 1 (e.g. when V is a constant), then the nested condition (c) can be removed, but the convergence (38) may no longer be stable, and the convergence (39) may no longer be mixing (see the remarks in p.59 and p. 64 of [START_REF] Hall | Martingale limit theory and its application[END_REF] for a triangular array). Lemma 3.1 can be extended in a clear way to two-sided martingale arrays {(S n,i , F n,i ) : -∞ < i < ∞}; for a version using conditions and norming in terms of i X 2 n,i , see Theorem 3.6 of [START_REF] Hall | Martingale limit theory and its application[END_REF] (p.77). Proof of Lemma 3.1. We will see that Lemma 3.1 can be obtained from the corresponding result for a triangular array of martingales. Let k n be positive integers increasing to ∞ such that

E(S n,∞ -S n,kn ) 2 = E i>kn X 2 n,i → 0. Then V 2 n,kn := kn i=1 E(X 2 n,i |F n,i-1 ) → V 2 in probability since V 2 n,∞ -V 2 n,kn = i>kn E(X 2 n,i |F n,i-1 ) → 0 in L 1 .
Clearly, by condition (b), the conditional Lindeberg condition for the triangular array {(X n,i , F n,i ) : Since S n,∞ -S n,kn → 0 in L 2 and hence in probability, (40) implies (38). As V 2 n,∞ -V 2 n,kn → 0 in probability (in fact in L 1 ), when V > 0 a.s. we have V 2 n,∞ /V 2 n,kn → 1 in probability. Therefore (41) implies (39). Lemma 3.1 is well suited for studying the rate of convergence of a martingale, as shown in the following Corollary 3.2 Let {(S i , F i ) : i ≥ 0} be a martingale defined on a probability space (Ω, F , P ), with S 0 = 0 and sup i≥1 ES 2 i < ∞. Let X i = S i -S i-1 , i ≥ 1 be the martingale differences, S ∞ = lim i→∞ S i be the a.s. limit of (S i ), and let

0 ≤ i ≤ k n } holds,
v 2 n = E(S ∞ -S n ) 2 = E ∞ i=n+1 X 2 i .
Suppose that v n > 0 and that:

(a) the conditional variance converges in probability: for a real random variable V ∈ [0, ∞),

V 2 n := 1 v 2 n ∞ i=n+1 E(X 2 i |F i-1 ) → V 2 in probability; (42)
(b) the conditional Lindeberg condition holds :

∀ε > 0, 1 v 2 n ∞ i=n+1 E(X 2 i 1 |X i |>εvn |F i-1 ) → 0 in probability. (43)
Then S ∞ -S n v n → V G in distribution, ( 44
)
where G is a Gaussian variable with law N(0, 1) and independent of V ; if additionally V = 0 a.s., then

S ∞ -S n V n → G in distribution. (45)
Moreover, the convergence in (44) is stable, and the convergence in (45) is mixing.

Proof. Corollary 3.2 is an immediate consequence of Lemma 3.1 applied to X n,i = X n+i for i ≥ 1, F n,i = F n+i for i ≥ 0, and X n,0 = 0.

Proof of Theorem 1.1. We can use Corollary 3.2 with 1/ W -W n 2 for the norming, but, since W -W n 2 ∼ σn -(d-2)/4 , we prefer to use the more explicit norming n (d-2)/4 , together with the spirit of the proof of Corollary 3.2. So we rely on the decomposition

n d-2 4 (W -W n ) = n d-2 4 ∞ k=n D k+1 , (46)
where

D k+1 = W k+1 -W k , k ≥ n,
forms a sequence of martingale differences. To prove Theorem 1.1, by (46) and Lemma 3. 6)), and X n,0 = 0, it suffices to prove that : (a) the following convergence about the conditional variance holds :

1 applied to X n,i = n d-2 4 D n+i for i ≥ 1, F n,i = G n+i for i ≥ 0 (recall (
s 2 n := n d-2 2 k≥n E k D 2 k+1 → σ 2 1 W 2 in probability, (47)
where

E k (•) = E(•|G k ) denotes the conditional expectation given G k ;
(b) the following Lindeberg condition holds :

∀ǫ > 0, n d-2 2 k≥n E k D 2 k+1 1 {n d-2 4 |D k+1 |>ε} → 0 in probability. (48)
Actually, by Lemma 3.1, from (47) and (48) we conclude that ( 12) and ( 13) hold with the norming 1/W n in (13) replaced by 1/W . As W n /W → 1 a.s. (and thus in probability), we can change the factor 1/W to 1/W n without changing the convergence in distribution.

To show the convergence (47) of the conditional variance, we will prove in the next section the following Lemma 3.3 There exists β 0 > 0 such that for |β| < β 0 and σ 1 from (30), we have, as n → ∞,

E(W n -W ) 4 -→ 0, (49)
Es 4 n -σ 4 1 EW 4 n -→ 0, (50)
E(s 2 n W 2 n ) -σ 2 1 EW 4 n -→ 0. (51)
It follows from Lemma 3.3 that

E(s 2 n -σ 2 1 EW 2 n ) 2 = Es 4 n -2σ 2 1 E(s 2 n W 2 n ) + σ 4 1 EW 4 n -→ 0. Therefore, s 2 n -σ 2 1 W 2 n → 0 in L 2 . As W 2 n → W 2 in L 2 , it follows that s 2 n → σ 2 1 W 2 in L 2 .
We thus obtain (47).

To show Lindeberg's condition (48), we will prove the following convergence rate of ED 4 k+1 .

Lemma 3.4 For any q > 1, when |β| > 0 is small enough, we have

ED 4 k+1 = O(k -d/q ), k ≥ 1. ( 52
)
The proof of the Lemma is postponed to the end of the paper. Note that a sufficient condition for Lindeberg's condition (48) to hold is clearly

n (d-2) k≥n E k D 4 k+1 → 0 in probability, which is implied by n (d-2) k≥n ED 4 k+1 → 0. (53)
By Lemma 3.4, when |β| > 0 is small enough, the left-hand side of (53) is smaller than c n d-1-d/q for some constant c > 0, which tends to 0 by taking 1 < q < d/(d -1), so that Lindeberg's condition (48) holds. (In fact one can check that it suffices to take |β| < β 0 with β 0 > 0 determined in Lemma 4.1.) This ends the proof of Theorem 1.1, using Lemmas 3.3 and 3.4 whose proofs will be given in the next section.

Remark 3.5 The convergence (13) can also be proved using the decomposition

n d-2 4 (W -W n ) W n = n d-2 4 W n ∞ k=n D k+1 ,
where (n

d-2 4 D k+1 /W n , k ≥ n)
remains a sequence of martingale differences. But proving (12) requires a different route.

We end this section with a warm-up calculation: we recover the value of EW 2 n and (18) from the martingale decomposition (46), i.e., using the conditional variance. This calculation is instructive and it will be useful in the forthcoming computations. Write for short

h k (S) = k-1 i=0 [βη(i, S i ) -λ(β)].
Since W k+1 -W k = P e h k (S) (e βη(k,S k )-λ(β) -1), (W k+1 -W k ) 2 = P ⊗2 e h k (S) (e βη(k,S k )-λ(β) -1)e h k ( S) (e βη(k, Sk )-λ(β) -1) , using Fubini's theorem we have

E k D 2 k+1 = κ 2 P ⊗2 e h k (S) e h k ( S) 1 {S k = Sk } (54) with κ 2 = κ 2 (β) = e λ 2 -1. ( 55
)
We compute s 2 n from its definition in (47):

s 2 n = κ 2 n d-2 2 k≥n P ⊗2 e h k (S) e h k ( S) 1 S k = Sk . (56)
Observe that

1 {S k = Sk } = e λ 2 (N k+1 -N k ) -1 κ 2 . ( 57
)
We now check that, with s 2 n from (56) and σ 2 be defined by ( 28), Es 2 n → σ 2 . This is a remake of [START_REF] Huang | Convergence rates for a supercritical branching process in a random environment[END_REF], but, as we will see, from a different route. By Fubini we have

Es 2 n = κ 2 n d-2 2 k≥n P ⊗2 Ee h k (S) e h k ( S) 1 S k = Sk = n d-2 2 P ⊗2 k≥n e λ 2 N k κ 2 1 S k = Sk = n d-2 2 P ⊗2 k≥n e λ 2 N k (e λ 2 (N k+1 -N k ) -1) (by (57)) = n d-2 2 P ⊗2 (e λ 2 N∞ -e λ 2 Nn ) (telescopic sum) = n d-2 2 P ⊗2 [P ⊗2 e λ 2 (N∞-Nn) -1 |F n )e λ 2 Nn ] = n d-2 2 P ⊗2 [F (S n -Sn )e λ 2 Nn ],
where we can express

F (x) = P ⊗2 0,x (e λ 2 N∞ -1) = P ⊗2 0,x ((e λ 2 N∞ -1)1 τ <∞ ) = P ⊗2 0,x (τ < ∞)P ⊗2 0,0 (e λ 2 N∞ -1) = G(x) G(0) (E(W 2 ) -1). (58)
Therefore, by the same argument as in [START_REF] Nguyen | Stochastic order and disordered systems[END_REF], we derive from the last 2 formulas,

Es 2 n = Var(W ) G(0) P ⊗2 [n d-2 2 G(S n -Sn )e λ 2 Nn ] → K d G(0) Var(W )E 1 |Z| d-2 × P ⊗2 e λ 2 N∞ = K d Var(W )(1 -π d )Z d × EW 2 which is equal to σ 2 .

Proof of the instrumental lemmas

In this section we give the proofs of Lemmas 3.3 and 3.4.

Proof of Lemma 3.3. We first give the proof of (49). For 4 independent paths S (i) , i = 1, . . . 4, we need to count the number of intersections of 2, 3 or four of them. Denote by E 4 the set of elements of (Z d ) 4 with all 4 elements equal, by E 3 the set of those with 3 equal lattice sites and a different fourth, by E 2,2 the set of those with 2 pairs of equal lattice sites but the two are different, and by E 2,0 the set of those with one pair being equal and different from the two other ones. Let A be the corresponding set of indices, A = {4, 3, (2, 2), (2, 0)}, and define

λ m = λ(mβ) -mλ(β), m ≥ 2, N (a) n,k = k-1 t=n 1 {(S (1) t ,S (2)
t ,S

t ,S

t )∈Ea} , a ∈ A,

Σ n,k = λ 4 N (4) n,k + λ 3 N (3) n,k + 2λ 2 N (2,2) n,k + λ 2 N (2,0) n,k , 0 ≤ n ≤ k.
Then it is elementary to check that

E exp{ 4 i=1 h n (S (i) )} = exp Σ 0,n , EW 4 n = P ⊗4 e Σ 0,n . (59)
For all pair S, S of paths and all 0 ≤ n ≤ k, put N n,k (S, S) = k-1 i=n 1 S i = S i . Note that, for all a ∈ A, N

n,k ≤ 1≤i<j≤4 N n,k (S (i) , S (j) ), (a)
and that λ m ց 0 as |β| ց 0. Thus, taking |β| > 0 small, we have uniform integrability of (W 4 n ) n , and then sup

n EW 4 n = EW 4 < ∞. (61)
This gives (49).

We next give the proof of (50). We compute

Es 4 n = κ 2 2 n d-2 k,ℓ≥n P ⊗4 Ee h k (S (1) )+h k (S (2) )+h ℓ (S (3) )+h ℓ (S (4) ) 1 S (1) k =S (2) k 1 S (3) ℓ =S (4) ℓ = T < (n) + T = (n) + T > (n),
where

T = (n) [resp. T < (n), resp. T > (n)] is the contribution in the sum of the terms with k = ℓ, [resp. k < ℓ, resp. k > ℓ]. By symmetry, T < (n) = T > (n). We calculate T < (n): T < (n) = κ 2 2 n d-2 P ⊗4 ℓ>k≥n e Σ 0,k +λ 2 N k,ℓ (S (3) ,S (4) ) 1 S (1) k =S (2) k 1 S (3) ℓ =S (4) ℓ = κ 2 n d-2 P ⊗4 k≥n e Σ 0,k 1 S (1) k =S (2) k ℓ>k e λ 2 N k,ℓ (S (3) ,S (4) ) κ 2 1 S (3) ℓ =S (4) ℓ (57) = κ 2 n d-2 P ⊗4 k≥n e Σ 0,k 1 S (1) k =S (2) k ℓ>k e λ 2 N k,ℓ (S (3) ,S (4) ) e λ 2 N ℓ,ℓ+1 (S (3) ,S (4) ) -1 = κ 2 n d-2 P ⊗4 k≥n e Σ 0,k 1 S (1) k =S (2) k ℓ>k e λ 2 N k,ℓ+1 (S (3) ,S (4) ) -e λ 2 N k,ℓ (S (3) ,S (4) ) = κ 2 n d-2 P ⊗4 k≥n e Σ 0,k 1 S (1) k =S (2) k e λ 2 N k,∞ (S (3) ,S (4) ) -e λ 2 N k,k+1 (S (3) ,S (4) ) = κ 2 n d-2 P ⊗4 k≥n e Σ 0,k 1 S (1) k =S (2) k e λ 2 N k,∞ (S (3) ,S (4) ) -1 -T = (n) = κ 2 n d-2 P ⊗4 k≥n e Σ 0,k 1 S (1) k =S (2) k P ⊗2 e λ 2 N k,∞ (S (3) ,S (4) ) -1 F k -T = (n) (58) = Var(W ) G(0) κ 2 × n d-2 P ⊗4 k≥n e Σ 0,k 1 S (1) k =S (2) k G(S (3)
k -S

k ) -T = (n) = T < 1 (n) -T = (n) , (4)
which serves also as the definition of T

< 1 (n). For 0 ≤ m ≤ n define T < 2 (m, n) = Var(W ) G(0) κ 2 × n d-2 k≥n P ⊗4 e Σ 0,m +λ 2 N k-m,k (S (1) ,S (2) ) 1 S (1) k =S (2) k G(S (3)
k -S

k ) .

It is sufficient to study the limit of T < 2 (m, n) for large n, m, since we will prove the following Lemma 4.1 Let β 0 > 0 be such that, for some ǫ > 0 and all |β| < β 0 , sup n P ⊗4 [e (d+ǫ)Σ 0,n ] < ∞. Then,

lim n→∞ E|T = (n)| = lim m→∞ lim sup n→∞ E T < 1 (n) -T < 2 (m, n) = 0. Remark 4.2
The occurrence of the term N k-m,k may be surprising. It is reminiscent of a similar phenomenon in the local limit theorem for polymers [START_REF] Sinai | A remark concerning random walks with random potentials Fund[END_REF][START_REF] Vargas | A Local limit theorem for directed polymers in random media: the continuous and the discrete case[END_REF]. The constraint S

k = S

(2) k makes likely intersections between S (1) and S (2) just before time k, whereas it is likely that S (3) and S (4) are far from them and far apart one from another.

With Lemma 4.1 we continue our proof. To analyse T < 2 (m, n) we condition on the vectors

S (1,2) m,k = S (1,2) = (S (i)
t ; i = 1, 2, t = 0, . . . m -1, and t = k), and S (3,4) = (S

t ; i = 3, 4, t = 0, . . . m -1), and use the independence of the paths S (i) :

T < 2 (m, n) = Var(W ) G(0) κ 2 n d-2 k≥n P ⊗4 e Σ 0,m 1 S (1) k =S (2) k
P ⊗2 e λ 2 N k-m,k (S (1) ,S (2) ) S (1,2) × P ⊗2 G(S

k -S

k ) S (3,4) ∼ Var(W )K P ⊗4 e Σ 0,m 1 S (1) k =S

(2) k P ⊗2 e λ 2 N k-m,k (S (1) ,S (2) ) S (1,2) (62)

as n → ∞, since for fixed m, k d-2

2 P ⊗2 G(S (3)
k -S

k ) S (3,4) → K d Z d uniformly as k → ∞ by ( 23), [START_REF] Lawler | Intersections of Random Walks[END_REF], [START_REF] Lawler | Random Walk: A Modern Introduction[END_REF]. Now, on the event {S → e -λ 2 C d P ⊗4 e Σ 0,m P ⊗2 e λ 2 N 0,m 1 d -2

.

Collecting all this we conclude that the limit lim n→∞ T < 2 (m, n) exists, and further, that the limits lim m→∞ lim n→∞ T < 2 (m, n) and lim n→∞ T < (n) exist and are equal. Finally, lim n→∞

Es 4 n = 2 lim n→∞ T < (n) (62) = 2 K d Z d C d (1-π d ) d -2 κ 2 e -λ 2 × E(W 4 )E(W 2 )Var(W ) = σ 4 1 E(W 4 ). ( 66
)
This concludes the proof of (50).

We then give the proof of (51). We estimate the cross term

Es 2 n W 2 n = κ 2 n d-2 2
k≥n P ⊗4 e Σ 0,n +λ 2 N n,k (S (1) ,S (2) ) 1 S (1)

k =S (2) k = κ 2 n d-2 2 k≥n P ⊗4 e Σ 0,n 1 S (1) k =S (2) k
P ⊗2 e λ 2 N n,k (S (1) ,S (2) ) S 1,2 n,k , and we proceed as in (63), (64). On the event {S

(1)

k = S (2)
k } {S

2 PP-λ 2 k d 2 PP∼ e -λ 2 k d 2 PP

 222 , i = 1, 2}, we have by time-reversal and Markov property,P ⊗2 e λ 2 N k-m,k (S (1) ,S (2) ) S (1,2) = e -λ 2 P ⊗2 e λ 2 N 0,m S (i) k-m = s (i) m-1 , i = 1, 2 → e -λ 2 P ⊗2 e λ 2 N 0,m ,(63)as k → ∞. Thus, for fixed m and k → ∞, it holdsk d ⊗4 e Σ 0,m 1 S (1) ⊗2 e λ 2 N k-m,k S (1,2) ∼ e ⊗4 e Σ 0,m 1 S (1) ⊗2 e λ 2 N 0,m ⊗4 e Σ 0,m P ⊗2 1 S (1) ⊗2 e λ 2 N 0,m → e -λ 2 C d P ⊗4 e Σ 0,m P ⊗2 e λ 2 N 0,m ,(64)by the local limit theorem (see e.g. [24, Theorem 1.2.1, p.14]), withC d = 2 d/4π d/2 . (65)It follows that the sum in (62) can be estimated, for n → ∞, by n d-2 k≥n of (62) ∼ e -λ 2 P ⊗4 e Σ 0,m C d P ⊗2 e λ 2 N 0,m n d

n- 1 2 P 2 n 2 × κ 2 e -λ 2 ×P≤

 12222 , i = 1, 2}, we have by time-reversal and Markov property,P ⊗2 e λ 2 N n,k (S (1) ,S (2) ) S (1,2) n,k = e -λ 2 P ⊗2 e λ 2 N 0,k-n S (i) k-n = s (i) n-1 , i = 1, 2 → e -λ 2 P ⊗2 e λ 2 N 0,∞ , as k → ∞, and uniformly in k ≥ n as n → ∞, k d ⊗4 e Σ 0,n +λ 2 N n,k (S (1) ,S (2) ) 1 S (1) k =S (2) k → e -λ 2 EW 2 C d EW 4 . = κ 2 e -λ 2 EW 2 C d EW 4 lim n→∞ n EW 4 EW 2 ,which ends the proof of (51).We now give the Proof of Lemma 4.1. For the first limit, we put q = (d + ǫ)/(d -1 + ǫ) and we estimate ET = ⊗4 e (d+ǫ)Σ 0,k 1/(d+ǫ) P ⊗2 [S Cn d-2 k≥n k -d/q

  instead of the Euclidean norm |x|, and the elementary inequality x 1 /d ≤ |x| ≤ x 1 valid for all x ∈ Z d .

Acknowledgements. The work is supported in parts by CNRS (UMR 7599 Probabilités et Modèles Aléatoires et UMR 6205 Laboratoire de Mathématique de Bretagne Atlantique), the National Natural Science Foundation of China (Grants no. 11571052 and no. 11401590) and the U.S. National Science Foundation (Grant no. NSF PHY11-25915).

with C ∈ (0, ∞) a constant, by the local limit theorem. Being of order n -ǫ/(d+ǫ) , the last term vanishes as n → ∞. We now prove the second limit, using arguments which are similar to the ones above. Since the difference is non-negative, the norm

k -S

k )

with C = Var(W ) G(0) κ 2 , by conditioning on the paths at times t = 1, . . . m and t = k. The event

t | ≥ m 1/4 ; 1 ≤ i = j ≤ 4, t = m, and t = k except for (i, j) = (1, 2) has probability larger than 1-cm -d/4 for some constant c ∈ (0, ∞), by the local limit theorem. On this event, by transience, the intersections between the paths between times m and km essentially come, when m and k are large, from those of S (1) and S (2) between times km and m. Precisely, (1) , S (2) )

(note that the event in the previous line means that there are no intersections of S (i) and S (j) between times m and k for i ≥ 3 or j ≥ 3, and no intersections of S (1) and S (2) between times m and km). Under our integrability condition (61), this is enough to imply that lim m→∞ lim sup

Plugging this in (67) and using the fact that n d-2 k≥n P ⊗4 [e Σ 0,m +λ 2 N k-m,k (S (1) ,S (2) ) 1 S (1) k =S

(2) k

] has a finite limit, we conclude that the right-hand side of (67) vanishes as n → ∞.

Proof of Lemma 3.4. With the notation introduced in the beginning of this section, we have

Using Fubini's theorem and independence, we obtain

k ,S

k ,S

k ,S

k ,S

where γ 4 = E(e βη(0,0)-λ(β) -1) 4 and γ 2 = (E(e βη(0,0)-λ(β) -1) 2 ) 2 . Notice that (S

k , S

k , S

k , S

k ) ∈ E 2,2 if and only if one of the following cases occurs: (a) S

k . Therefore by symmetry, we obtain

.

Hence, using Hölder's inequality, for p, q > 1 with 1/p + 1/q = 1, we have ED 4 k+1 ≤ (γ 4 + 3γ 2 )(P ⊗4 e pΣ 0,k ) 1/p (P ⊗2 (S

k )) 2/q .

By the local limit theorem, P ⊗2 (S

). Thus taking |β| > 0 small enough such that P ⊗4 e pΣ 0,k < ∞, we see that Eq. ( 52) holds. This ends the proof of Lemma 3.4.