FLUCTUATION EXPONENTS AND LARGE DEVIATIONS FOR DIRECTED POLYMERS IN A RANDOM ENVIRONMENT

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1 FLUCTUATION EXPONENTS AND LARGE DEVIATIONS FOR DIRECTED POLYMERS IN A RANDOM ENVIRONMENT PHILIPPE CARMONA AND YUEYUN HU Abstract. For the model of directed polymers in a gaussian random environment introduced by Imbrie and Spencer, we establish: a scaling inequality between the volume exponent and the fluctuation exponent of the free energy; a Large Deviations Principle for the end position of the polymer under the Gibbs measure; a relationship between the volume exponent and the rate function of the Large Deviations Principle.. Introduction Let us first describe the model of directed polymers in a random environment introduced by Imbrie and Spencer 2.The random media is a family (g(k, x), k, x Z d ) of independent identically distributed random variables defined on a probability space (Ω, F, P). Let S n be the simple symmetric random walk in Z d. Let P x be the probability measure of (S n ) n N starting at x Z d and let E x be the corresponding expectation. The object of our study is the Gibbs measure (n) defined by (n) def f = E 0 f(s)e βh n(g,s), H n (g, γ) def = Z n where Ω n is the set of nearest neighbour paths of length n: n g(i, γ(i)). (.) Ω n def = { γ :, n Z d, γ(k) γ(k ) =, k = 2,..., n }, and Z n is the partition function: i= (.2) Z n = Z n (g, β) = E 0 e βh n(g,s). Date: January 24, Mathematics Subject Classification. Primary 60K37. Key words and phrases. Random Environment; Directed Polymers; Large Deviations; Concentration of Measure.

2 2 P. CARMONA AND Y. HU In other words, for a given realization g(ω) of the environment, the Gibbs measure gives to a polymer γ having an energy H n (g, γ) at temperature T = β, a weight proportional to eβhn(g,γ). Dating back to the pioneer work of Imbrie and Spencer 2, the situation in dimension d 3 is well understood for small β > 0. There exists some constant β 0 > 0 such that for β < β 0, S n is diffusive under (n). Theorem.. (Imbrie and Spencer 2, Bolthausen 3, Sinai 8, Albeverio and Zhou ) If d 3, then there exists β 0 = β 0 (d) > 0 such that for β < β 0, Sn 2 (n) (.3) n P a.s.. It is believed in many physics papers (see e.g. 6, 3) that for small dimension d =, 2 and β small enough S n 2 (n) behaves as n 2ζ for some ζ >. However no rigorous proof has been given yet. 2 The aim of this paper is to establish rigorous results when the random media is given by i.i.d. N (0, ) gaussian random variables. Following Piza 6 we define the volume exponent (.4) ζ def { } (n) = inf α > 0 : ( max k n S k n α ) in P probability, and the fluctuation exponent of the free energy (.5) χ = sup { α > 0 : Var(log Z n ) n 2α for large enough n }. We shall establish a scaling inequality similar to the one obtained by Piza 6 in a different framework. He works with a polymer model more related to oriented percolation, where furthermore the potentials g are assumed non positive. Theorem.2. For all d and β 0, χ dζ. 2 The following Large Deviations result may seem at first sight highly uncorrelated to these two exponents.

3 3 Theorem.3. The distribution of S n under the Gibbs measure (n) n satisfy a Large Deviations Principle for closed sets with a (deterministic) good rate function K β. For every closed set C of, d, (n) lim sup n n log Sn n C inf K β(θ) P a.s.. θ C We shall prove the existence of a pointwise rate function Proposition.4. There exists a deterministic nonnegative convex function I β : (, ) d R +, such that I β (0) = 0. for θ (, ) d Q d, n log (n) (Sn =nθ) Iβ (θ), P a.s.. where we take the limit along n such that P 0 (S n = nθ) > 0. I β (θ) K β (θ) However, there is a close relationship between the volume exponent and the shape of the rate function. Theorem.5. Assume that for some δ, C > 0 the rate function satisfies: (.6) I β (θ) C θ δ (θ (, ) d ). Then, the volume exponent satisfies: (.7) ζ 2δ. In particular, if lim n E log Z n(β) = 2 β2, which is the case if d 3 and β is small enough 4, then ζ 3/4. When (S n ) is replaced by a discrete time Brownian motion and g(, ) by a gaussian fields, Petermann 5 showed ζ 3/5 in the onedimensional case and Mejane 4 showed ζ 3/4 for all d. But it does not seem clear to obtain a invariance principle between these two models. Acknowledgements. We are very much indebted to M. Talagrand s First Course on spin glasses 20 from which we learned how to use effectively concentration of measure and integration by parts. Furthermore, we would like to stress the fact that while in spin glasses the covariance structure of the energies of configurations is influenced by the exchangeability of the individuals spins, in the polymer model it is influenced by the Markov property of the underlying random walk.

4 4 P. CARMONA AND Y. HU 2. Notations and basic tools In addition to the partition function Z n, we define the partition function starting from x: Z n (x) = Z n (x, g) def = E n x e βh n (g,s), H n (g, γ) = g(i, γ(i)), and the point to point partition function i= Z n (x, y) = Z n (x, y, g) = E x e βh n (g,s) (Sn=y). This function is strictly positive only if there exists a nearest neighbour path of length n connecting x to y, i.e. P x S n = y > 0. We shall note this x n y. Eventually, let θ n denote the time shift of order n on the environment: Our first tool is the (θ n g)(k, x) = g(k + n, x) (x Z d, k ). Lemma 2. (Markov Property). For every integers n, m and every x, y Z d, we have: (2.) Z n+m (x) = y Z n (x, y, g)z m (y, θ n g) (2.2) Z n+m (x, z) = y Z n (x, y, g)z m (y, z, θ n g). Proof. The two identities are proved similarly, with the help of the Markov property for the random walk S n. Indeed, Therefore, H n+m (g, S) = n+m i= g(i, S i ) = H n (g, S) + H m (θ n g, S n+. ). Z n+m (x, g) = E x e βh n(g,s) e βhm(θng,s n+.) = E x e βh n (g,s) Z m (S n, θ n g) = y E x e βh n(g,s) (Sn =y) Zm (y, θ n g) = y Z n (x, y, g)z m (y, θ n g). Let us recall the concentration of measure property of Gaussian processes (see Ibragimov and al. ).

5 5 Proposition 2.2. Consider a function F on R M and assume that its Lipschitz constant is at most A, i.e. F (x) F (y) A x y (x, y R M ), where x denotes the euclidean norm of x. Then if g = (g,..., g M ) are i.i.d. N (0, ) we have P ( F (g) E F (g) u) exp( u2 ) (u > 0). 2A2 Following Talagrand 9, we apply this Proposition to partition functions. Proposition Define for n integer p n (β) = E log Z n n. Then, for any u > 0, and x n 0, (2.3) (2.4) ( ) P n log Z n p n (β) u exp( nu2 2β 2 ) P ( log Z n (0, x) E log Z n (0, x) u) exp( u2 2nβ 2 ). 2. Let ν > 2. Then almost surely there exists n 0 such that for every n n 0 : (2.5) (2.6) log Z n (β) np n (β) n ν log Z n (0, x) E log Z n (0, x) n ν (x Z d, x n 0). 3. Let ν >. There exists a constant C = C(ν, β) > 0 such that: 2 E exp(n ν (log Z n (0, x) E log Z n (0, x))) C (n, x n 0). Proof.. We consider the function F : R M R defined by n F (z) = log E 0 exp(β z i,si ) = log E 0 exp(β = log E 0 e β as.z i= n z i,x (Si =x)) i= x i, where a γ is the vector with coordinates: a γ i,x = (γ(i)=x) ( i n, x i). Cauchy-Schwarz inequality yields a γ.z a γ.z a γ z z = n z z.

6 6 P. CARMONA AND Y. HU Therefore, F has Lispschitz constant at most A = β n, and we obtain the concentration of measure inequality (2.3). The inequality (2.4) is obtained in the same way. 2. The second part of the Proposition is proved by introducing the events: A n = x Z d : x n{ log Z n (0, x) E log Z n (0, x) n ν } { log Z n (β) np n (β) n ν }. Then, P (A n ) c d n d exp( n2ν ), 2β 2 n P (A n) < + and we conclude by Borel Cantelli s Lemma. 3. The third part is a straightforward upper-bound using the concentration inequality E e n ν (log Z n (0,x) Elog Z n (0,x)) ( ) = P e n ν (log Z n(0,x) Elog Z n(0,x)) v dv P ( log Z n (0, x) E log Z n (0, x) n ν log v) dv exp ( (log ) v)2 n 2ν dv 2β 2 exp ( C 4 (log v) 2) dv = C 5 < +. Eventually let us recall the well known result (see e.g. Talagrand 20) Lemma 2.4. Let (g(i)) i N be a family of N (0, ) random variables, not necessarily independent. Then, E max g(i) 2 log N. i N Proof. This short proof was given by M. Talagrand during the Ecole d été de Saint-Flour, but was not incorporated in the notes 20. We let g = max i N g(i). Then for all real number β, by Jensen s inequality N Ne β2 2 = E e βg(i) E e βg exp(βe g ). i= Therefore, β2 +log N βe 2 g and optimizing for β = 2 log N yields the desired result.

7 7 3. A scaling inequality involving the volume and the fluctuation exponents The following lemma is elementary, but nevertheless gives useful lower bounds on the variance: Lemma 3.. Let X L 2 (Ω, F, P) and assume that G,..., G k,... is a sequence of mutually independent sub-σ-fields of F. We have ( ) Var(X) Var E X G k. k= Lemma 3.2. Denote by g = d N (0, ) a standard real-valued Gaussian variable. There exists some constant c β > 0 such that for all u, v 0, we have ( ) Cov log(u + e βg ), log(v + e βg ) ( c β max 0, ı (u,v ), ) uv ı (u>,v>). Proof. Denote by f(u, v) the above covariance. Since f(u, v) = Cov ( log(+ u eβg ), log( + v eβg ) ), we get lim u f(u, v) = 0. Let g be an independent copy of g, we have (3.) 2 f u v (u, v) = Cov( u + e, ) βg v + e βg = E (u + e βg )(v + e βg ) (u + e βg )(v + e βeg ) e βeg e βg = E (u + e βg )(v + e βg )(v + e βeg ) (e βeg e βg )(u + e βeg ) = E (u + e βg )(u + e βeg )(v + e βg )(v + e βeg ) (e βeg e βg )e βeg = E (u + e βg )(u + e βeg )(v + e βg )(v + e βeg ) = 2 E (e βeg e βg ) 2, (u + e βg )(u + e βeg )(v + e βg )(v + e βeg ) where the last equality is obtained by interchanging g and g. Remark that for x, y 0, f(x, y) = x f f(u, y)du = u Going back to (3.), we remark that inf 0 u 2,0 v 2 x du 2 f u v (u, v) = c β <, y dv 2 f (u, v) 0. u v

8 8 P. CARMONA AND Y. HU and for all u, v, f (u, v) = u v 2u 2 v E (e βeg e βg ) 2 2 ( + u eβg )( + u eβeg )( + v eβg )( + v eβeg ) From the above estimates, the desired conclusion follows (the constant c β may be smaller). c βu 2 v 2. Proof of Theorem.2. Applying Lemma 3. with {G k, k } = { σ(g(j, x)), j n, x Z d}, we get ( ) ( ) Var log Z n (β) Var E log Z n (β) g(j, x). j n,x Z d Fix j n and x Z d. Denote by D n (g, S) def = exp We have ( β ) n g(i, S i). Z n (β) = E 0 Dn (g, S) (Sj x) +E0 Dj (g, S) (Sj =x) e βg(j,x) Z n j (x, g θ j ) It follows that log Z n (β) = log ) (Y +e βg(j,x) +log Z n j (x, g θ j )E 0 Dj (g, S) (Sj =x), with (here we can only consider those j and x such that P 0 S j = x > 0) Y def E 0 Dn (g, S) (Sj x) = Z n j (x, g θ j )E 0 Dj (g, S) (Sj =x) is independent of g(j, x). Since Z n j (x, g θ j )E 0 Dj (g, S) (Sj =x) is also independent of g(j, x), we get ( ) Var E log Z n (β) g(j, x) = E P (Y dy) y>0 = (( c β max c β 4 E ( P (Y dy )P (Y dy 2 ) Cov Y = Var (E log(y + e βg(j,x) ) g(j, x) ) log(y + e βg(j,x) ) E log(y + e βg(j,x) ) 2 ) 2, P (Y ) (E ) 2) Y (Y >) 2, ) log(y + e βg(j,x) ), log(y 2 + e βg(j,x) ) where the first inequality is due to Lemma 3.2. Recalling the definition of Y, we remark that Y = (Sj =x) (n) e βg(j,x) (Sj =x) (n) e βg(j,x) (Sj =x) (n),

9 9 which in view of Lemma 3. imply that Var(log Z n ) c ( β E ( e βg(j,x) (Sj =x) (n))) 2. 4 j n,x Z d Pick up ζ > ζ. Cauchy-Schwarz inequality implies that (E ( e βg(j,x) (Sj =x) (n))) 2 j n, x n ζ ( j n, x n ζ E ( e βg(j,x) (Sj=x) (n))) 2 where ( n ζ d ( n ζ d j n, x n ζ E j n, x n ζ E j n j n, x n ζ ( e M n (g) (Sj =x) (n))) 2 ( e M n (g) (Mn(g) 0) (Sj =x) (n))) 2 ( ) 2 = n ζ d E e Mn(g) (Mn (g) 0) ( Sj n ζ ) (n), M n (g) def = max g(j, x). j n, x n The standard extreme value theory 7 implies that if then, N n def = {(j, x) : j n, x n} = (n + )(2n + ) d n d+ M n (g) 2 log Nn, almost surely. We introduce the event { A n = 2 log Nn M n 2 } 2 log N n. 2 We have P (A n ). By definition of the exponent ζ, if X n = ( sup k n S k n ζ ) (n), then X n in probability and 0 X n. Therefore, X n in L (P), and E X n An. So, there exists n 0 such that for n n 0, E X n An 2. Consequently, for n n 0 and j n,

10 0 P. CARMONA AND Y. HU (n) E e Mn(g) (Mn (g) 0) ( S j n ζ ) E e Mn(g) X n An Hence exp( 2 2 log N n )E X n An 2 exp( 2 2 log N n ). 2 log Nn, Var(log Z n ) c β 6 n ζ d e 4 which implies that 2χ ζ d for all ζ > ζ, ending the proof. 4. Large Deviations Principle The first step in the proof of Theorem.3 is the Proposition 4.. For every λ R d, we have the convergence lim n + n log e λ.sn (n) = φβ (λ) Pa.s. and in L (P). The function φ β : R d R is convex nonnegative, φ β (0) = 0, and for every permutation σ: φ β (λ,..., λ d ) = φ β (±λ σ(),... ± λ σ(d) ). Proof. Fix λ R d and define Zn(x, λ g) = E x e λ.(s n x) e P β n i= g(i,s i), and Zn λ = Zn(0, λ g). From the Markov property of the random walk S we infer that e λ.s n (n) Zn+m λ = Zn λ (Sn =x) Z λ e λ.s n (n) m(x, θ n g). x Jensen s inequality applied to the concave function log yields e λ.s n (n) (Sn=x) log Z λ n+m log Z λ n + x e λ.s n (n) log Z λ m(x, θ n g). The independence between the σ-fields σ(g(k, x), k n, x Z d ) and σ(g(k, x), k > n, x Z d ) implies that E log Zn+m λ E log Z λ e λs n (n) (Sn=x) n + E E log Z λ x e λ.s n (n) m(x, θ n g) = E log Zn λ + E log Z λ e λs n (n) (Sn =x) m E e λs n (n) = E log Z λ n + E log Z λ m. x

11 Consequently, the sequence n E log Z λ n is superadditive and we can consider the limit: φ β (λ) def = lim n E log e λ.s n (n). We can apply the concentration of measure property (see Proposition 2.2) to see that this convergence is also an almost sure convergence. To prove that this convergence also holds in L (P) we only need to show that the sequence log e n λs (n) n is uniformly integrable. This is true, because this sequence is bounded by c d λ. The properties of convexity, positivity and symmetry of φ β are inherited from the functions λ log E e λs n (n). For example, by convexity n of the exponential: n E log e λ.s (n) n E λ.s n (n) = 0. Proof of Theorem.3. This is Gärtner-Ellis Theorem (see e.g. Dembo and Zeitouni Theorem , or den Hollander section V.2 0). We can obtain a weak LDP if we are able to prove that the function λ φ β (λ) is differentiable on R d. We now state a Law of Large Numbers in dimension d =.We say that (S n ) n N is a nearest neighbour random walk on Z if for a constant p it is a Markov chain with transition probabilities: p(x, x + ) = p = p(x, x ) (x Z). Proposition 4.2. Assume that the function φ β is differentiable at λ R. Then the nearest neighbour random walk on Z with mean tanh λ, i.e. p = eλ, satisfies the law of large numbers: 2chλ n S n (n) φ β(λ) almost surely. This very weak law of large numbers is not a surprise for the symmetric random walk, since we have then S n (n) = d S n (n) and thus E S n (n) = 0. Proof. The function φ β is the limit of the convex C functions f n (λ) = log e n (n). λs n Therefore, since φβ is differentiable at λ (see Lemma 4.3),

12 2 P. CARMONA AND Y. HU almost surely φ β(λ) = lim f n(λ) = lim n Sn e λs n (n) e λs n (n). Define ψ(λ) = log cosh(λ). Then M λ n = e λs n nψ(λ) is a martingale and under the new probability E λ f(s k, k n) def = E 0 f(sk, k n)e λsn nψ(λ) the nearest neighbour random walk S has mean tanh(λ). Let us denote by S (λ) this walk ; then almost surely φ β(λ) = lim S (λ) (n) n, n which is the desired result. Lemma 4.3. Assume the sequence f n of real valued, differentiable, convex functions converge on an open interval I to a function f. Then f is convex, and for every x such that f is differentiable at x, f n(x) converges to f (x). Proof. The convexity of f is obvious. For a < b in I we have f n (b) f n (a) b a Therefore, since f n(a) fn(b) fn(a) b a lim sup f n(a) Letting b decrease to a, we obtain Similarly, since f n(b) f n(b) f n (a) b a Letting a increase to b yields: f(b) f(a) b a we obtain: f(b) f(a) b a lim sup f n(a) f +(a). we obtain: lim inf f n(b) f n(b) f n (a) b a lim inf f n(b) f (b). Combining these inequalities yields for any x such that f is differentiable at x, and therefore lim inf f n(x) f (x) = f +(x) lim sup f n(x), lim f n(x) = f (x)....

13 5. A relationship between the volume exponent and the shape of the rate function It is easy to adapt the arguments of the proof of Proposition 4. to establish the convergence of the free energy: 3 (5.) p(β) def = lim n n log Z n P a.s. and in L (P). Indeed, the sequence n p n (β) = n E log Z n is superadditive so p(β) = lim p n (β) is well defined. The concentration of measure property (Proposition 2.3) implies that n log Z n p n (β) 0 a.s and in L. See also Comets, Shiga and Yoshida 5 where they studied general environments g by using martingale arguments. Our aim now is to show that there exists an exponential rate of convergence of the point to point partition function when we follow a given direction. Indeed, from the Markov Property (Lemma 2.) we get Z n+m (0, x + y, g) Z n (0, x, g)z m (x, x + y, θ n g) (x, y Z d ). Taking the expectation of the logarithm, we get, since θ n g is distributed as g, (5.2) (5.3) E log Z n+m (0, x + y) E log Z n (0, x) + E log Z m (x, x + y) = E log Z n (0, x) + E log Z m (0, y) Fix e = (k, z), k, z Z d. the sequence n a n = E log Z nk (0, nz) is superadditive : a n+m a n + a m. hence, we can consider the limit (5.4) f β (e) = f β (k, z) def = lim n n E log Z nk(0, nz) = sup n n E log Z nk(0, nz). Following Piza 6 and Alexander 2, we can define f β (e) for e = (k, z) having rational coordinates by restricting our sequence n a n to the n such that ne = (nk, nz) has integer coordinates. We have defined on Q + Q d a function f β such that: f β is homogeneous: f β (λe) = λf β (e) for λ Q +. f β is super additive: f β (e + e ) f β (e) + f β (e ). We can then conclude by Lemma.5 of Alexander 2 that f β may be extended to a uniformly continuous function on R + R d, homogeneous, superadditive and concave. Remark 5.. Owing to the concentration of measure phenomenon, the convergence f β (k, z) def = lim log Z n nk(0, nz) also holds almost surely and in L (P).

14 4 P. CARMONA AND Y. HU Remark 5.2. We need to be very careful when defining f β. Indeed if e = (k, z) and P 0 S nk = nz = 0, i.e. there is no path of length nk connecting 0 to nz, then Z nk (0, nz) = 0 and f β (e) =. Therefore, if z > k we have f β (k, z) = and when z k, we shall restrict the sequence n E log Z nk (0, nz) to the integers n such that P 0 S nk = nz > 0. If this is the case, we do not have integrability problems preventing us from establishing (5.2). Indeed, Jensen s inequality implies that: Z nk (0, nz) P 0 S nk = nze 0 e βh nk (g,s) S nk = nz Therefore, P 0 S nk = nz exp(βe 0 H nk (g, S) S nk = nz). n E log Z nk(0, nz) n log P 0S nk = nz + β n E E 0H nk (g, S) S nk = nz = n log P 0S nk = nz. and this entails, thanks to the local central limit theorem for ordinary random walks (see e.g. Lawler 8), the lower bound, C d = 2(d/2π) d/2 : z 2 f β (k, z) C d k. Let us now establish some useful properties of the function f β. Proposition 5.3. () f β (, 0) = sup θ (,) d f β (, θ). (2) f β (, 0) = p(β). (3) Fix ν >. Then, for m large enough 2 m E log Z m p(β) m E log Z m + m ν and almost surely, for m large enough m log Z m m ν p(β) m log Z m + m ν (4) Similarly, for m large enough, 2m E log Z 2m(0, 0) f β (, 0) 2m E log Z 2m(0, 0) + m ν. (5) β 2 2 C d θ 2 f β (, θ) C d θ 2. Proof.. The Markov Property (Lemma 2.) implies that for x Q d Z 2n (0, 0, g) Z n (0, nx, g)z n (nx, 0, θ n g).

15 Therefore, (5.5) E Z 2n (0, 0) E log Z n (0, nx) + E log Z n (nx, 0) = 2E log Z n (0, nx). Taking limits along n such that P 0 S n = x > 0 and nx Z d, we get: f β (2, 0) = 2f β (, 0) 2f β (, x). 2. Fix ν > 2 and let ɛ = n ν. Then, 5 E log Z n ɛ log E Zɛ n (Jensens inequality) ɛ log E Z n (0, x) ɛ (since 0 < ɛ < ) x n = ɛ log E e ɛ(log Zn(0,x) Elog Zn(0,x)) e x n ɛ log C x n e ɛelog Z n(0,x) ɛelog Zn(0,x) (by Proposition 2.3) ɛ log ( C(2n + ) d e ɛ/2elog Z 2n(0,0) ) (by inequality 5.5) C n ν + C 2 (log n)n ν + 2 E log Z 2n(0, 0). We conclude by taking limits. 3. The second line follows from the first and from the concentration of measure property. By construction m E log Z m p(β). Therefore, we only need to establish that for m large enough p(β) m E log Z m + m ν. Fix a large m and a small 0 < ɛ = ɛ(m) < whose value will be determined later. Define h ɛ (n) def = log E Z ɛ n ɛe log Z n. Observe that by the Markov property and the concavity of x x ɛ, ( ) ɛ Zn+k ɛ = Z n (0, x, g)z k (x, θ n g) x x Z ɛ n(0, x, g)z ɛ k(x, θ n g).

16 6 P. CARMONA AND Y. HU Hence, by independence, h ɛ (n + k) log E Zn(0, ɛ x, g)e Zk(x, ɛ θ n g) x n log ( (2n) d E Z ɛ ne Z ɛ k ) = h ɛ (n) + h ɛ (k) + d log(2n). Thanks to Hammersley s general subaditive theorem 9, the following limit exists and satisfies: h ɛ (n) h(ɛ) = lim h ɛ(m) n + n m + C log m m. Recall that h ɛ (n) ɛe log Z n. Hence, h(ɛ) ɛp(β). We now choose ɛ = m ν with ν > /2. We obtain that h ɛ (m) = log E e ɛ(log Z m Elog Z m ) + ɛe log Z m Therefore, C + ɛe log Z m (by Proposition 2.3). p(β) h(ɛ) h ɛ(m) + C log m ɛ mɛ mɛ m E log Z m + C m ν log m, proving the desired result for ν > ν. 4. Since the proof is really similar to the preceding one, it has been omitted. 5. Only the left inequality remains to be proved. Thanks again to the concavity of the logarithm: n E log Z nk(0, nz) n log E Z nk(0, nz) = ( ) n log β2 nk e 2 P0 S nk = nz. We obtain from the local central limit theorem that, f β (k, z) kβ2 2 C z 2 d k. Proof of Proposition.4. Define (5.6) I β (θ) = p(β) f β (, θ). Then, thanks to Proposition 5.3, the function I β and I β (0) = 0. It remains to show that: I β (θ) K β (θ). is positive, convex

17 Indeed, if θ has rational coordinate, then C = {θ} is a closed set and (n) Sn log n C = log Z n(0, nθ). Z n Therefore, if we let n go to with the restrictions that nθ Z d and P 0 S n = nθ > 0, we get: (n) n log Sn n C f β (, θ) p(β) = I β (θ). On the other hand, the Large Deviations Property of Theorem.3 implies that lim sup (n) n log Sn n C K β (θ), 7 so we obtain I β (θ) K β (θ). Proof of Theorem.5. Fix ν > and γ such that + δ(γ ) > ν. 2 Using the concentration of measure inequality as in the proof of Proposition 2.3, it is easy to show that almost surely for n large enough, k n and z k 0: log (n) (Sk =z) E log (n) (Sk =z) 2n ν. Assume that k is even, so that 0 k 0. Then, Markov property implies (n) Z k (0, z)z n k (z, θ k g) (Sk =z) = x Z k(0, x)z n k (x, θ k g) Z k(0, z) Z k (0, 0) Since Z n k (z, θ k g) d = Z n k (0, θ k g), we obtain Z n k (z, θ k g) Z n k (0, θ k g). E log (n) (Sk =z) E log Z k (0, z) E log Z k (0, 0) f β (k, z) E log Z k (0, 0) (by (5.4)) f β (k, z) k f β (, 0) + k ν (by Proposition 5.3) = ki β (z/k) + k ν. Hence, if z n γ, we get almost surely for large enough n,

18 8 P. CARMONA AND Y. HU log (n) (Sk =z) kiβ (z/k) + k ν + 2n ν Thus, Ck z/k δ + 3n ν Ck δ n γδ + 3n ν (since z n γ ) Cn δ+γδ + 3n ν (since k n and δ > ) C 2 n δ+γδ. (n) ( Sk n γ ) C3 n d e C 2n δ+γδ. If k is odd, then (n) (n) ( Sk n γ ) ( Sk n ) γ and we obtain the same type of upper bound. This upper bound implies that (n) ( k n, Sk n ) γ k n ( Sk n γ ) Therefore, P alsmost surely, ( max k n S k n γ ) (n), (n) C3 n d+ e C 2n δ+γδ 0 and γ ζ. We conclude by letting ν 2 and γ 2δ. Assume now that p(β) = β 2 /2. By Proposition 5.3 this implies that I β (θ) C d θ 2. We can apply the preceding result with δ = 2 and obtain ζ 3/4. References. S. Albeverio and X.Y. Zhou, A martingale approach to directed polymers in a random environment, Journal of Theoretical Probability 9 (995), no., K.S. Alexander, Approximation of subadditive functions and convergence rates in limiting shape results, The Annals of Probability 28 (997), no., E. Bolthausen, A note on the diffusion of directed polymers in a random environment, Communications in Mathematical Physics 23 (989), Ph. Carmona and Y. Hu, On the partition function of a directed polymer in a Gaussian random environment, Probab. Th. Rel. Fields 24 (2002) F. Comets, T., Shiga and N. Yoshida, Directed polymers in random environment: Path localization and strong disorder. Bernoulli (to appear). 6. D.A. Huse and C.L. Henley, Pinning and roughening of domain walls in Ising systems due to random impurities, Phys. Rev. Lett. 54 (985), A. Dembo and O. Zeitouni, Large deviations techniques and applications, Applications of Mathematics, no. 38, Springer, G.F. Lawler, Intersections of random walks, Probability and its applications, Birkhauser, 996.

19 9. J.M. Hammersley, Generalization of the fundamental theorem on subadditive functions, Proc. Cambridge Philosophical Society 58 (962), F. den Hollander, Large deviations, Fields Institute Monographs, vol. 4, American Mathematical Society, I.A. Ibragimov, V. Sudakov, and B.S. Tsirelson, Norms of Gaussian sample functions, Proceedings of the Third Japan-USSR Symposium on Probability (Tashkent), Lecture Notes in Mathematics, vol. 550, 975, pp J.Z. Imbrie and T. Spencer, Diffusion of directed polymers in a random environment, Journal of Statistical Physics 52 (988), M. Kardar, Roughening by impurities at finite temperatures, Phys. Rev. Lett. 55 (985), O. Mejane, Upper bound of a volume exponent for directed polymers in a random environment. (preprint 2002). 5. M. Petermann, Sperdiffusivity of directed polymers in random environment, Ph.D thesis, Univ. Zürich, M.S.T. Piza, Directed polymers in a random environment : Some results on fluctuations, Journal of Statistical Physics 89 (997), no. 3/4, S.I. Resnick, Extreme values, regular variation and point processes, Applied Probability, vol. 4, Springer, Y.G. Sinai, A remark concerning random walks with random potentials. Fund. Math. 47 (995), no. 2, M. Talagrand, The Sherrington-Kirkpatrick model : a challenge to mathematicians, Probability Theory and Related Fields 0 (998), , Ecole d été de probabilité de saint-flour, Lecture Notes in Mathematics, ch. A First course on Spin Glasses, Springer, Philippe Carmona, Laboratoire de Mathématiques Jean Leray, Université de Nantes, 2 Rue de la Houssinière, BP 92208, F Nantes Cedex 03 France address: philippe.carmona@math.univ-nantes.fr Yueyun Hu, Laboratoire de Probabilités et Modèles Aléatoires, CNRS UMR 7599, Université Paris VI, case 88, 4 place Jussieu Paris Cedex 05 France address: hu@ccr.jussieu.fr 9