Upper bound of a volume exponent for directed polymers in a random environment

Upper bound of a volume exponent for directed polymers in a random environment

Olivier Mejane

Laboratoire de statistiques et probabilités, Université Paul Sabatier, 118, route de Narbonne, 31062 Toulouse, France Received 18 March 2003; received in revised form 16 October 2003; accepted 21 October 2003

Available online 13 February 2004

Abstract

We consider the model of directed polymers in a random environment introduced by Petermann: the random walk is R^d-valued and has independent N(0, I_d)-increments, and the random media is a stationary centered Gaussian process (g(k, x), k1, x∈R^d)with covariance matrix cov(g(i, x), g(j, y))=δ_ij(x−y),whereis a bounded integrable function onR^d.For this model, we establish an upper bound of the volume exponent in all dimensionsd.

2004 Elsevier SAS. All rights reserved.

Résumé

On considère le modèle de polymères dirigés en environnement aléatoire introduit par Petermann : la marche aléatoire sous- jacente est à valeurs dansR^d, ses incréments sont des variables indépendantes de loiN(0, I_d), et le milieu aléatoire est un processus gaussien stationnaire centré(g(k, x), k1, x∈R^d)de matrice de covariance cov(g(i, x), g(j, y))=δ_ij(x−y), oùest une fonction bornée intégrable surR^d. Pour ce modèle, nous établissons une majoration de l’exposant de volume, pour toute dimension d.

2004 Elsevier SAS. All rights reserved.

MSC: 60K37; 60G42; 82D30

Keywords: Directed polymers in random environment; Gaussian environment

1. Introduction

The model of directed polymers in a random environment was introduced by Imbrie and Spencer [7]. We focus here on a particular model studied by Petermann [8] in his thesis: let (Sn)n0 be a random walk in R^d starting from the origin, with independentN(0, Id)-increments, defined on a probability space(Ω,F,P), and let g=(g(k, x), k1, x∈R^d)be a stationary centered Gaussian process with covariance matrix

cov

g(i, x), g(j, y)

=δij(x−y),

E-mail address: olivier.mejane@lsp.ups-tlse.fr (O. Mejane).

0246-0203/$ – see front matter 2004 Elsevier SAS. All rights reserved.

doi:10.1016/j.anihpb.2003.10.007

whereis a bounded integrable function onR^d.We suppose that this random mediagis defined on a probability space(Ω^g,G, P ), where(Gn)_n0is the natural filtration:

Gn=σ

g(k, x), 1kn, x∈R^d

forn1 (G0 being the trivialσ-algebra). We denote by E (respectivelyE) the expectation with respect toP (respectivelyP). We define the Gibbs measure.⁽ⁿ⁾by:

f⁽ⁿ⁾= 1 ZnE

f (S₁, . . . , S_n)e^{βnk=1g(k,Sk)}

for any bounded functionf on(R^d)ⁿ, whereβ >0 is a fixed parameter andZ_nis the partition function:

Z_n=E

e^{βnk=1g(k,Sk)} .

Following Piza [9] we define the volume exponent ξ=inf

α >0: 1_{max_kn|S_k|n^α}⁽ⁿ⁾ →

n→∞1 in P-probability .

Here and in the sequel, |x| =max_1id|xi| for any x =(x1, . . . , xd)∈R^d. Petermann obtained a result of superdiffusivity in dimension one, in the particular case where(x)=_2λ¹e^−λ|x| for someλ >0: he proved that ξ3/5 for allβ >0 (for another result of superdiffusivity, see [6]).

Our main result gives on the contrary an upper bound for the volume exponent, in all dimensions:

∀d1, ∀β >0 ξ3

4. (1)

This paper is organized as follows:

• In Section 2, we first extend exponential inequalities concerning independent Gaussian variables, proved by Carmona and Hu [1], to the case of a stationary Gaussian process. Then, following Comets, Shiga and Yoshida [2], we combine these inequalities with martingale methods and obtain a concentration inequality.

• In Section 3, we obtain an upper bound for ξ when we consider only the value of the walkS at time n, and not the maximal one beforen. In fact we prove a stronger result, namely a large deviation principle for (1Sn/n^α∈.⁽ⁿ⁾, n1)whenα >3/4. This result and its proof are an adaptation of the works of Comets and Yoshida on a continuous model of directed polymers [3].

• In Section 4, we establish (1).

• Appendix A is devoted to the proof of Lemma 2.4, used in Section 2, which gives a large deviation estimate for a sum of martingale-differences. It is a slight extension of a result of Lesigne and Volný [5, Theorem 3.2].

2. Preliminary: a concentration inequality 2.1. Exponential inequalities

Lemma 2.1. Let(g(x), x∈R^d)be a family of Gaussian centered random variables with common varianceσ²>0.

We fixq, β >0,(x₁, . . . , x_n)∈(R^d)ⁿand(λ₁, . . . , λ_n)inRⁿ. Then for any probability measureµonR^d: e^−β

2σ2

2 qE

e^{βni=1λig(xi)}

Re^βg(x)µ(dx)q

e^β

2σ2 2

q+n i=1|λi|².

The proof is identical with the one made by Carmona and Hu in a discrete framework (µis the sum of Dirac masses), and is therefore omitted.

Lemma 2.2. Let (g(x), x ∈R^d) be a centered Gaussian process with covariance matrix cov(g(x), g(y))= (x−y). Letσ²=(0), and letµ be a probability measure on R^d. Then for allβ >0, there are constants c1=c1(β, σ²) >0 andc2=c2(β, σ²) >0 such that:

−c₁

R^d

(x−y) µ(dx) µ(dy)E

log

R

e^βg(x)−β

2σ2 2 µ(dx)

−c₂

R^d

(x−y) µ(dx) µ(dy).

In particular,

−c1σ²E

log

R^d

e^βg(x)−β

2σ2 2 µ(dx)

0.

Proof. Let{B_x(t), t0}x∈R^d be the family of centered Gaussian processes such that E

Bx(t)By(s)

=inf(s, t)(x−y), withB_x(0)=0 for allx∈R^d. Define

X(t)=

R^d

M_x(t) µ(dx), t0,

whereM_x(t)=e^{βBx(t )−β2σ2t /2}. SincedM_x(t)=βM_x(t) dB_x(t), one has

dMx, Myt=β²Mx(t)My(t) dBx, Byt=β²e^{β(Bx(t )+By(t ))−β2σ2t}(x−y) dt, anddX, Xt=

R^dβ²e^{β(Bx(t )+By(t ))−β2σ2t}(x−y) µ(dx) µ(dy) dt.Thus, by Ito’s formula, E(logX₁)= −β²

2

R^d

µ(dx) µ(dy) (x−y)

1

0

E

e^{β(Bx(t )+By(t ))−β2σ2t} X²_t

dt.

By Lemma 2.1, we have for allt: e^−β2σ2tE

e^{β(Bx(t )+By(t ))−β2σ2t} X_t²

=E

e^{β(Bx(t )+By(t ))} (

Re^{βBx(t )}µ(dx))²

e^8β2σ2t. Hence:

−e^8β2σ2−1 16σ²

R^d

(x−y) µ(dx) µ(dy)E(logX₁)−1−e^−β2σ2 2σ²

R^d

(x−y) µ(dx) µ(dy),

which concludes the proof sinceX1

=d

R^de^βg(x)−β

2σ2

2 µ(dx). 2 2.2. A concentration result

Proposition 2.3. Letν >1/2. Forn∈N,jnandfna nonnegative bounded function, such thatE(fn(Sj)) >0.

We noteW_n,j=E(f_n(S_j)e^βnk=1g(k,S_k)). Then fornn₀(β, ν), PlogWn,j−E(logWn,j)n^ν

exp

−1

4n^(2ν−1)/3

.

Proof. We use the following lemma, whose proof is postponed to Appendix A.

Lemma 2.4. Let(Xⁱ_n, 1in)be a martingale difference sequence and letM_n=_n

i=1X_nⁱ. Suppose that there existsK >0 such thatE(e^|Xin|)Kfor alliandn. Then for anyν >1/2, and fornn₀(K, ν),

P

|Mn|> n^ν

exp

−1

4n^(2ν−1)/3

.

• We first assume thatf_n>0.

To apply the Lemma 2.4, we defineXⁱ_n,j=E(logW_n,j|Gi)−E(logW_n,j|Gi−1)so that log(Wn,j)−E(logWn,j)=

n

i=1

X_n,jⁱ .

It is sufficient to prove that there existsK >0 such thatE(e^|Xin,j|)Kfor alliand(n, j ).

For this, we introduce:

eⁱ_n,j=f_n(S_j)exp

1kn, k=i

βg(k, S_k)

, W_n,jⁱ =E eⁱ_n,j

.

W_n,jⁱ >0 since we assumed that fn >0. If Ei is the conditional expectation with respect to Gi, then Ei(logW_n,jⁱ )=Ei−1(logW_n,jⁱ ),so that:

X_n,jⁱ =Ei

logY_n,jⁱ

−Ei−1

logY_n,jⁱ

, (2)

with

Y_n,jⁱ =e^−β2/2W_n,j W_n,jⁱ =

R^d

e^{βg(i,x)−β2/2}µⁱ_n,j(dx), (3)

µⁱ_n,j being the random probability measure:

µⁱ_n,j(dx)= 1 W_n,jⁱ E

eⁱ_n,j|Si=x

P(Si∈dx).

Since µⁱ_n,j is measurable with respect to Gn,i =σ (g(k, x),1kn, k=i, x∈R^d), we deduce from Lemma 2.2 that there exists a constantc=c(β) >0, which does not depend on(n, j, i), such that:

−cE

log

R^d

e^{βg(i,x)−β2/2}µⁱ_n,j(dx)|Gn,i

0,

and sinceGi−1⊂Gn,i, we obtain:

0−E_i−1

logY_n,jⁱ

c. (4)

Thus we deduce from (2) and (4) that for allθ∈R E

e^{θ Xin,j}

e^cθ+E

e^{θ Ei(logYn,ji )} withθ⁺:=max(θ ,0). By Jensen’s inequality,

e^{θ Ei(logYn,ji )}Ei

Y_n,jⁱ θ

so that E

e^{θ Xin,j}

e^cθ+E Y_n,jⁱ θ

=e^cθ+E E

Y_n,jⁱ θ

|Gn,i

.

Assume now that θ ∈ {−1,1}, hence in both cases, the function x →x^θ is convex; using (3), we obtain (Y_n,jⁱ )^θ

R^de^{θ (βg(i,x)−β2/2)}µⁱ_n,j(dx),so that:

E Y_n,jⁱ θ

|Gn,i

R^d

E

e^{θ (βg(i,x)−β2/2)}|Gn,i

µⁱ_n,j(dx)=

R^d

E

e^{θ (βg(i,x)−β2/2)}

µⁱ_n,j(dx)

=E

e^{θ (βg(1,0)−β2/2)} ,

using thatg(i, x)is independent fromGn,i, and is distributed asg(1,0)for alliandx. We conclude that for allnand 1i,jn,

E e^|Xin,j|

E

e^Xin,j +E

e^−Xin,j

K:=e^c+e^β2.

• In the general case wheref_n0, we introduceh_n=f_n+δ for some 0< δ <1. The first part of the proof applies toh_n: notingW_n,j^δ =E(h_n(S_j)e^βnk=1g(k,S_k)), it remains to show that logW_n,j^δ −E(logW_n,j^δ )^P-a.s.

δ→→0

logW_n,j−E(logW_n,j). Sincef_nis bounded by some constantC_n>0, the following inequality holds for all 0< δ <1: logW_n,j logW_n,j^δ log((C_n+1)Z_n). Since 0ElogZ_nlogEZ_n=nβ²(0)/2<∞, the conclusion follows from dominated convergence. 2

Corollary 2.5. Letν >1/2. Let us fix a sequence of Borel sets(B(j, n), n1, j n). ThenP-almost surely, there existsn0such that for everynn0, everyjn,

log1Sj∈B(j,n)⁽ⁿ⁾−E

log1Sj∈B(j,n)⁽ⁿ⁾2n^ν.

Proof. Let us write A_n,j = {|logE(f_n(S_j)e^βnk=1g(k,S_k))−E[logE(f_n(S_j)e^βnk=1g(k,S_k))]|n^ν}. Proposi- tion 2.3 implies that

P

jn

A_n,j

nexp

−1

4n^(2ν−1)/3

.

Hence, by Borel–Cantelli,P-almost-surely there existsn₀such that for everynn₀and everyj n:

logE

f_n(S_j)e^{βnk=1g(k,Sk)}

−E logE

f_n(S_j)e^{βnk=1g(k,Sk)}n^ν. Then one applies this result tofn(x)=1x∈B(j,n)and tofn(x)=1. 2

3. A first result

In this section, we prove that a large deviation principle holdsP-almost surely for the sequence of measures (1S_n/n^α∈.⁽ⁿ⁾, n1)ifα >3/4. This was first proved by Comets and Yoshida [3, Theorem 2.4.4], for a model of directed polymers in which the random walk is replaced by a Brownian motion and the environment is given by a Poisson random measure onR₊×R^d.

Theorem 3.1. Letα >3/4. Then a large deviation principle for(1Sn/n^α∈.⁽ⁿ⁾, n1)holdsP-a.s., with the rate functionI (λ)= λ²/2 and the speedn^2α−1,.denoting the Euclidean norm onR^d. In particular, for allε >0,

nlim→∞− 1

n^2α−1log1Snεn^α(n)=ε²

2 P-a.s.

Remark 3.2. In particular this result implies that for allα >3/4, 1_|Sn|n^{α(n) P}-a.s.

n→∞→ 0. (5)

Proof. Let us fixλ∈R^d,n1, and then introduce the following martingale:

M_p^λ,n=

e^{λ.Sp−pλ2/2} ifpn, e^{λ.Sn−nλ2/2} ifp > n,

withx.y denoting the scalar product between two vectorsx andy inR^d. If Q^λ,n is the probability defined by Girsanov’s change associated to this positive martingale, Girsanov’s formula ensures that, underQ^λ,n, the process (S˜_p:=S_p−λ(p∧n), p1)has the same distribution asSunderP. Therefore:

Z_n e^λ.Sn(n)

=e^nλ2/2E

M_n^λe^{βnk=1g(k,Sk)}

=e^nλ2/2E

e^{βnk=1gλ,n(k,Sk+kλ)}

=e^nλ2/2E e^β

n

k=1g^λ,n(k,Sk) ,

where we denote byg^λ,nthe translated environment g^λ,n(k, x):=g

k, x+λ(k∧n) .

By stationarity, this environment has the same distribution as(g(k, x), k1, x∈R^d), hence E

e^{βnk=1gλ,n(k,Sk)} d

=E

e^{βnk=1g(k,Sk)} , thus

Elog e^λ.Sn(n)

=nλ²/2. (6)

Now let us fixα >3/4. Withn^α−1λinstead ofλ, (6) gives Elog

e^nα−1λ.Sn(n)

=n^2α−1λ²/2. (7)

Let us definef_n(x)=e^nα−1λ.x. This function is positive andE(f_n(S_n)e^βnk=1g(k,Sk)) <∞, so that the result of Corollary 2.5 is still true withf_n(x)instead of1x∈B(j,n). Since 2α−1>1/2, this implies

nlim→∞

1 n^2α−1

log

e^nα−1λ.Sn(n)

−Elog

e^nα−1λ.Sn(n)

=0 P-a.s. (8)

From (7) and (8), we get:

nlim→∞

1 n^2α−1log

e^nα−1λ.Sn(n)

= λ²/2 P-a.s.

Let us defineh_n(λ)=_n2α−1¹ loge^{nα−1λ.Sn(n)}andh(λ)= λ²/2. From what we proved, we deduce the existence ofA⊂Ω^g, with P (A)=1, on whichhn(λ)→h(λ)for allλ∈Q^d. Now we show that onAthe convergence actually holds for allλ∈R^d, by using that the functionshnare convex andhis continuous. To this goal, we can reduce the proof to the cased=1. Indeed ifd 2, we fix(d−1)coordinates in Q^d−1and use thathn is still convex as a function of the last coordinate, and then repeat the process. So let us assumed=1 and fixλ∈R^∗. There exist two sequences(a_i, i0)and(b_i, i0)inQ^N that converge toλ,(a_i, i0)being increasing and(b_i, i0)decreasing. Let us fixi1 andn1. Sinceh_n is convex and satisfiesh_n(0)=0, the function x→h_n(x)/xis increasing. Hence the following inequalities hold:

h_n(a_i)

a_i h_n(λ)

λ h_n(b_i) b_i .

Sincehnconverges towardshonQ, it follows that h(ai)

ai lim inf

n→∞

hn(λ)

λ lim sup

n→∞

hn(λ)

λ h(bi) bi

.

The limit functionhbeing continuous, we obtain by lettingi→ +∞that the limit ofhn(λ)/λexists and is equal toh(λ)/λ, which proves thath_n(λ)→h(λ)for allλ∈R.

One then concludes by the Gärtner–Ellis–Baldi theorem (see [4]). 2

4. Upper bound of the volume exponent

We now extend the result (5) to the maximal deviation from the origin:

Theorem 4.1. For alld1 andα >3/4, 1_{maxkn|Sk|n^α}^{(n) P}-a.s.

n→∞→ 0.

Proof. We will use the following notations: forx∈R^dandr0,B(x, r)= {y∈R^d, |y−x|r}. Forα0 and j=(j₁, . . . , j_d)∈Z^d,B_j^α=B(j n^α, n^α). We will use the fact that the union of the balls(B_j^α, j ∈(2Z)^d\{0})form a partition ofR^d\B(0, n^α).

We first prove the following upper bound:

Proposition 4.2. Letn0 andkn. Then for anyj∈Z^dandα >0, E

log1Sk∈B_j^α(n)

−n^2α−1 2

d

i=1

(ji−εi)²,

whereεi=sgn(ji) (=0 ifji=0).

Proof. Let note a_k,j^α =E(1_Sk∈B^α

je^βni=1g(i,S_i)), so that 1Sk∈B^α_j⁽ⁿ⁾ =a_k,j^α /Z_n. Let be λ = ˜λ/k with λ˜_i = (ji−εi)n^α, 1id; then let us define the martingale

M_p^λ,k=

e^{λ.Sp−pλ2/2} ifpk, e^{λ.Sk−kλ2/2} ifp > k,

wherex.y denotes the usual scalar product inR^d andxthe associated euclidean norm. Under the probability Q^λ,kassociated to this martingale,(S_p)_p0has the law of the following shifted random walk underP:

S˜_p=S_p+ ˜λ p

k ∧1

.

It follows that:

a_k,j^α =E

e^{−k1(˜λ.Sk+˜λ}

2/2)1_S

k∈B_j^α−˜λe^β

n

i=1g(i,S˜ i)

, (9)

whereg(i, x)˜ =g(i, x+ ˜λ(i/k∧1)).

Now we notice that on the event{S_k∈B_j^α− ˜λ}, one hasλ.S˜ _k0: indeed if we writeS_k=(S_k¹, . . . , S_k^d), then for any 1id,|S_kⁱ −j_in^α+ ˜λ_i|n^α, hence:

• forji1,λ˜i=(ji−1)n^α0 and 0S_kⁱ 2n^α,

• forji−1,λ˜i=(ji+1)n^α0 and−2n^αS_kⁱ 0,

• forj_i=0,λ˜_i=0,

so that in all casesλ˜_iS_kⁱ 0 and thusλ.S˜ _k0. Therefore on the event{S_k∈B_j^α− ˜λ}, e^{−k1(˜λ.Sk+˜λ}

2/2)e^−˜λ

2

2n e⁻ⁿ

2α−1 2

d 1(ji−εi)², and (9) leads to:

a_k,j^α e⁻ⁿ

2α−1 2

d

1(ji−εi)²E 1_S

k∈B_j^α−˜λe^{βni=1g(i,S˜ i)} .

On the other hand,ZnE(1_S

k∈B_j^α−˜λe^{βni=1g(i,Si)}), and since by stationarity the environmentg˜ has the same distribution asg, it follows that for allj∈Z^d,

E

log1_Sk∈B^α

j⁽ⁿ⁾

−n^2α−1 2

d

i=1

(ji−εi)². 2

Letν >1/2. We deduce from Proposition 4.2 and from Corollary 2.5 (with B(k, n)=B_j^α) that,P-a.s., for nn₀,kn, and allj ∈Z^d:

log1Sk∈B_j^α(n)2n^ν−n^2α−1 2

d

i=1

(j_i−ε_i)².

So,P-a.s., fornn0,1|S_k|n^α(n)

j∈(2Z)^d\{0}e^2nν−n

2α−1 2

d

i=1(j_i−ε_i)²,and 1_{max_kn|S_k|n^α}⁽ⁿ⁾

n

k=1

1_|S_k|n^α(n)

j∈(2Z)^d\{0}

ne^2nν−n

2α−1 2

d

i=1(ji−εi)². But by symmetry, for anyC >0,

j∈(2Z)^d\{0}

e^{−Cdi=1(ji−εi)2}2d

j₁2

e^{−C(j1−1)2} d

i=2

j_i∈2Z

e^{−C(ji−εi)2}

and using that

j2e^−C(j−1)2

j1e^−Cj =₁₋^e−C_e−C,we conclude that,P-a.s., for some constantC(d) >0, and fornn0:

1_{max_kn|S_k|n^α}⁽ⁿ⁾C(d)ne^2nν e^{−n2α−1/2} 1−e^{−n2α−1/2}. Thus for allα >^ν+₂¹,P-a.s.,

1_{max_kn|S_k|n^α}⁽ⁿ⁾ →

n→∞0.

This is true for allν >1/2, which ends the proof. 2

Appendix A. Proof of Lemma 2.4

The beginning of the proof is exactly identical with the one made by Lesigne and Volný in [5, pp. 148–149].

Only the last ten lines differ.

Let us denote(Fnⁱ)_1inthe filtration of(Xⁱ_n)_1in. The hypothesis that it is a martingale difference sequence means that for eachi,Xⁱ_nisFnⁱ-measurable and, ifi2,E[X_nⁱ |Fnⁱ⁻¹] =0.

Let us fixa >0 and for 1indefine Y_nⁱ=Xⁱ_n1_|Xi

n|an^1/3−E X_nⁱ1_|Xi

n|an^1/3Fnⁱ⁻¹

and

Z_nⁱ =Xⁱ_n1_|Xi

n|>an^1/3−E Xⁱ_n1_|Xi

n|>an^1/3F_nⁱ⁻¹ ,

and then define M_n =_k

i=1Y_nⁱ and M_n =_k

i=1Z_nⁱ. Since (Xⁱ_n)_1in is a martingale difference sequence, (Y_nⁱ)_1inand(Zⁱ_n)_1inare martingale difference sequences andXⁱ_n=Y_nⁱ+Zⁱ_n(1in).

Let us fixt∈(0,1). For everyx >0, P

|Mn|> nx

P

|M_n|> nxt +P

|M_n|> nx(1−t)

. (A.1)

Since|Y_nⁱ|2an^1/3for 1in, Azuma’s inequality implies:

P

|M_n|> nxt

=P |M_n|

2an^1/3> nxt 2an^1/3

2 exp

−t²x² 8a²n^1/3

. (A.2)

To control the second term in (A.1), we notice thatE((M_n)²)=_n

i=1E(Z_nⁱ)². For each 1in, if we note F_nⁱ(x)=P(|Xⁱ_n|> x):

E Z_nⁱ2

=E Xⁱ_n1_|Xi

n|>an^1/3

2

−E E

Xⁱ_n1_|Xi

n|>an^1/3Fnⁱ⁻¹

2

E

Xⁱ_n1_|Xi n|>an^1/3

2

= −

+∞

an^1/3

x²dF_nⁱ(x).

SinceEe^|Xin|K,F_nⁱ(x)Ke^−xfor allx0, hence:

−

+∞

an^1/3

x²dF_i(x)Ka²n^2/3e^−an1/3+2K

+∞

an^1/3

xe^−xdx=K

a²n^2/3+2an^1/3+2 e^−an1/3.

It follows thatE((M_n)²)nK(a²n^2/3+2an^1/3+2)e^−an1/3,and:

P

|M_n|> nx(1−t)

K

x²(1−t)²

a²n^−1/3+2an^−2/3+2n⁻¹

e^−an1/3. (A.3)

We choosea=¹₂(tx)^2/3so that^t2x2

8a² =a. From (A.1), (A.2) and (A.3), we deduce:

P

|Mn|> nx

2+ K

(1−t)²f (t, x, n)

exp

−1

2(tx)^2/3n^1/3

, (A.4)

withf (t, x, n)=¹₄t^4/3x^−2/3n^−1/3+t^2/3x^−4/3n^−2/3+2x⁻²n⁻¹. Now by takingx=n^ν−1, we have:

P

|M_n|> n^ν

=P

|M_n|> nx

2+ K

(1−t)²g(t, n)

exp

−1

2t^2/3x^2/3n^1/3

, (A.5)

with g(t, n)=f (t, n^ν−1, n)= ¹₄t^4/3n^{−(2ν−1)/3}+t^2/3n^{−2(2ν−1)/3}+2n^−(2ν−1). Now we fix ε >0 and choose t₀∈(0,1)such that 0<1−t₀^2/3< ε/2. (A.5) implies that:

P

|Mn|> n^ν exp

1

2(1−ε)n^(2ν−1)/3

2+ K

(1−t₀)²g(t0, n)

exp

−ε

4n^(2ν−1)/3

.

But, sinceν >1/2, (2+₍₁₋^K_t

0)²g(t₀, n))exp(−₄^εn^(2ν−1)/3) →

n→∞0. Therefore there existsn₀(ε) such that, for all nn0(ε),

P

|M_n|> n^ν

exp

−1

2(1−ε)n^(2ν−1)/3

. (A.6)

Whenε=1/2 this is exactly the statement of the Lemma 2.4.

References

[1] P. Carmona, Y. Hu, On the partition function of a directed polymer in a Gaussian random environment, Probab. Theory Related Fields 124 (3) (2002) 431–457.

[2] F. Comets, T. Shiga, N. Yoshida, Directed polymers in random environment: path localization and strong disorder, Bernoulli 9 (4) (2003) 705–723.

[3] F. Comets, N. Yoshida, Brownian directed polymers in random environment, Comm. Math. Phys. (2003), submitted for publication.

[4] A. Dembo, O. Zeitouni, Large Deviations Techniques and Applications, second ed., in: Applications of Mathematics, vol. 38, Springer- Verlag, New York, 1998.

[5] E. Lesigne, D. Volný, Large deviations for martingales, Stochastic Process. Appl. 96 (1) (2001) 143–159.

[6] C. Licea, C.M. Newman, M.S.T. Piza, Superdiffusivity in first-passage percolation, Probab. Theory Related Fields 106 (4) (1996) 559–591.

[7] J.Z. Imbrie, T. Spencer, Diffusion of directed polymers in a random environment, J. Statist. Phys. 52 (3–4) (1988) 609–626.

[8] M. Petermann, Superdiffusivity of directed polymers in random environment, Part of thesis, 2000.

[9] M. Piza, Directed polymers in a random environment: Some results on fluctuations, J. Statist. Phys. 89 (3–4) (1997) 581–603.