SYMMETRIC LANGEVIN SPIN GLASS DYNAMICS. By G. Ben Arous and A. Guionnet Ecole Normale Superieure and Université de Paris Sud

Transcription

1 The Annals of Probability 997, Vol. 5, No. 3, SYMMETRIC LANGEVIN SPIN GLASS DYNAMICS By G. Ben Arous and A. Guionnet Ecole Normale Superieure and Université de Paris Sud We study the asymptotic behavior of symmetric spin glass dynamics in the Sherrington Kirkpatrick model as proposed by Sompolinsky Zippelius. We prove that the averaged law of the empirical measure on the path space of these dynamics satisfies a large deviation upper bound in the high temperature regime. We study the rate function which governs this large deviation upper bound and prove that it achieves its minimum value at a unique probability measure Q which is not Markovian. We deduce an averaged and a quenched law of large numbers. We then study the evolution of the Gibbs measure of a spin glass under Sompolinsky Zippelius dynamics. We also prove a large deviation upper bound for the law of the empirical measure and describe the asymptotic behavior of a spin on path space under this dynamic in the high temperature regime.. Introduction. The Sherrington Kirkpatrick (S K) model is a mean field simplification of the spin glass model of Edwards Anderson. The behavior of its static characteristics, such as its partition function, has been intensively studied by physicists (see [] for a broad survey). There are few mathematical results available (except for [], [6], [9] and [7]). In [], it is argued that studying dynamics might be simpler since it avoids using the replica trick and the Parisi ansatz for symmetry breaking, which are yet to be put on firm ground. It seems that, in the physics literature, the first attempt to study the dynamics of S K is due to Sompolinsky and Zippelius (see [5]), who chose a Langevin dynamics scheme. In [3], we followed this strategy for asymmetric dynamics (which are not directly relevant to the study of statics for the S K model). We obtained there a full large deviation principle for path space empirical measure averaged on the Gaussian couplings (for short times or large temperatures). This large deviation principle enabled us to derive the so-called self-consistent limiting dynamics, which proved to be non-markovian. Here we want to attack the real problem, that is, symmetric dynamics. We prove only a strong large deviation upper bound with a good rate function. Minimizing this rate function gives a theorem on convergence to selfconsistent limiting dynamics, which we identify, though in a rather cryptic form. We can do this only in a short time or high temperature regime, and so this prevents us from drawing any conclusion for the behavior in large time, at fixed temperature, which would be a line of attack to study the equilibrium Received September 995; revised October 996. AMS 99 subject classifications. 6F, 6H, 6K35, 8C44, 8C3, 8C. Key words and phrases. Large deviations, interacting random processes, statistical mechanics, Langevin dynamics. 367

2 368 G. BEN AROUS AND A. GUIONNET measure. Weaker results concerning these dynamics are proved in [] for any time and temperature. To be more specific, let us recall that the S K Hamiltonian is given, for x = x x N N,by H N N J x = J ij x i x j N i j= where the randomness in the spin glass is here modeled by the J ij i j which are independent centered Gaussian random variables, and where J ij = J ji. The Gibbs probability measure one would like to study (for N large) is given by exp βh N J x α N dx Z N J where α = δ + δ and β is the inverse of temperature. Here Z N J is the partition function Z N J = exp βh N N J x x N If one replaces the hard spins + by continuous spins, that is, by spins taking values in R, or as we shall see in a bounded interval of R, and if one replaces the measure α = δ + δ by α dx = e U x / e U x dx dx, where U is, for instance, a double well potential on R, then the Langevin dynamics for this problem are given by () dx j t = db j t U x j t dt + β J ji x i t dt N i N where B is an N-dimensional Brownian motion. We want to understand the limiting behavior (for large N) of the law, say Pβ N J, of these randomly interacting diffusions given the initial law, say µ N. As in [3], we will only study bounded spins; that is, we will assume that U x is defined on a bounded interval A A and tends to infinity when x A sufficiently fast to insure our spins x j stay in the interval A A. However, we will not assume as in [3] that the the whole matrix J ij i j is made of i.i.d N random variables but rather assume the symmetry of couplings; that is, we will here suppose that the random matrix J ij i j is symmetric, that is, J ij = J ji. More precisely, we will suppose that under the diagonal, the J ij s are i.i.d N and N on the diagonal. Such a choice of covariance is nice from the technical point of view since it makes the law of the J ij s invariant by rotation. On the other hand, it does not interfere with the limit behavior of the spin glass. So, under this symmetry hypothesis, our dynamics () are reversible and their invariant measure is given by the Gibbs measure: { N } µ N J dx =exp βh N J x U x i N i= dx i i=

3 SYMMETRIC LANGEVIN SPIN GLASS DYNAMICS 369 Thus the symmetry hypothesis is crucial to understanding S K dynamics. On the other hand, this model is much more difficult to understand than the asymmetric one. Our first goal is to study the empirical measure ˆµ N = /N N i= δ x i on path space. There is no reason for this to be a simple problem, since, for fixed interaction J, the variables x x N are not exchangeable. So we first study the law of the empirical measure ˆµ N averaged on the interaction, leaving for a later work the study of J almost sure properties of this law. The main result of this paper is large deviation upper bounds for this averaged law in a large temperature (or short time) regime, which entails a propagation of chaos result, that is, a theorem on convergence to a probability measure on path space that we describe explicitly as the law of a non-markovian, highly nonlinear, solution of a stochastic differential equation (see Corollary 3.). The existence and uniqueness problems for this limit law are not obvious and are the analogue here of the existence and uniqueness problem for asymmetric spin glass dynamics as obtained in [3]. As in [3], we then deduce that the quenched law of the empirical measure converges exponentially fast to δ Q, which entails quenched laws of large numbers. We finally underline how our method can be used to study the evolution of the Gibbs measure µ N J under Sompolinski Zippelius dynamics and prove that, in the high temperature or short time regime, the quenched law of the empirical measure converges to the weak solution of a new nonlinear stochastic differential equation. The organization of the paper is as follows. In Section, we state and prove the strong large deviation upper bound. For more detail, see the following.. In Section., we introduce the rate function and state the strong large deviation upper bound (see Proposition. and Theorem.3).. In Section., we prove that the law of the path space dynamics averaged on the couplings is absolutely continuous with respect to the law of these dynamics with no couplings and show that its Radon Nikodym derivative is a function of the empirical measure. 3. In Section.3, we study the continuity properties of this density. 4. In Section.4, these continuity properties enable us to prove that the rate function is a good rate function in the short time or high temperature regime. 5. In Section.5, we prove the strong large deviation upper bound in the short time or high temperature regime by first proving an exponential tightness result and then a weak large deviation upper bound. In Section 3, we study the minima of the good rate function and prove that it achieves its minimum value at a unique probability measure, say Q. We describe Q as the unique solution of a fixed point problem in Theorem 3.4. This gives a propagation of chaos result stated in Corollary 3.3. In order to give a hint about what kind of result this approach leads to, let us state

4 37 G. BEN AROUS AND A. GUIONNET here Corollary 3.3.(ii): For any bounded continuous functions f f m on C T A +A [ ] m lim f x f m x m dpβ N J x = f i x dq x N where is the expectation over the Gaussian couplings. In Section 3., we characterize the minima of the good rate function. In Section 3., we reduce the problem of finding these minima to a fixed point problem and then we show that this fixed point problem has at most one solution. In Section 4, we apply our strategy to the stationary law of spin glass dynamics starting from the Gibbs measure. To this end, we need to suppose that β is small enough so that we are below the phase transition and that the free energy concentrates as proved by Talagrand (see [7]). Then, the study of the law of the empirical measure is reduced to that of the law of the empirical measure starting from the nonnormalized Gibbs measure Z N J µn J, which can be studied following the above procedure. We then describe the asymptotic behavior of the empirical measure.. Averaged and quenched large deviation upper bounds... Statement of the large deviation upper bound. We first make precise the setting of our model: let A be a strictly positive real and U be a C function on the interval A A such that U tends to infinity, when x A, sufficiently fast to insure that x lim x A ( y exp U y i= ) exp U z dz dy =+ For any number N of particles, any temperature (= /β) and J = J ij i j N R N N, we consider the following system β N J of interacting diffusions. For j N, β N J = dx j t = U x j t dt + db j t + β N J ji x i t dt N i= Law of x = µ N where B j j N is an N-dimensional Brownian motion and µ is a probability measure on A A which does not put mass on the boundary A +A. Under these assumptions, we recall Proposition. of [3]. Proposition.. For each J R N N, β N J has a unique weak solution and, almost surely, sup s T sup j N x j s does not reach A. In the following pages, we will focus on the evolution of this dynamical system until a time T and denote by Pβ N N J the weak solution of β J

5 SYMMETRIC LANGEVIN SPIN GLASS DYNAMICS 37 restricted to the sigma algebra T = σ x i s i N s T, and by P N the weak solution of N J restricted to T. Let W A T be the space of continuous functions from T into A A. Then Proposition. insures that Pβ N J is a probability measure on ( N. WT) A We now suppose that the J ij s are random and that their distribution is given by the following.. For any integer numbers i j, J ij = J ji.. If i<j, the J ij s are independent centered Gaussian variables with covariance. 3. The J ii s are independent centered Gaussian variables with covariance. They are also independent of the J ij i<j. We shall denote by γ the law of the J ij s and by expectation under γ. We have already noticed in [3] that P N β J is a measurable function of the J ij s. Further, we will be interested in the averaged law Q N β : Q N β = P N β J ω dγ ω The aim of this section is to prove that the law of the empirical measure under Q N β satisfies a large deviation upper bound, which entails a quenched large deviation upper bound. To this end, we first define the rate function H which governs this upper bound (see Proposition.). In order to define H, we need some notation and definitions that will also be useful later.. Let { / ( T = µ + WA T } U x s ds) dµ x < +. Let µ be a probability measure in. We denote by L µ WA T the space of the square integrable functions under µ. Hence L µ WA T is a Hilbert space with scalar product f g µ = gf dµ. 3. Let I be the identity on L µ WA T. 4. Let T be an integral operator on L µ WA T with kernel b T x y = T x t y t dt Then T is a symmetric nonnegative Hilbert Schmidt operator in L µ WA T [for any µ + WA T ]. 5. Let λ i be the eigenvalues of T in L µ WA T, and E i i N be an orthonormal basis of eigenvectors of T such that T E i = λ i E i. Since T is nonnegative, the λ i s are nonnegative so that we can define a symmetric positive Hilbert Schmidt operator log I + β T in L µ WA T by i N log I + β T E i = log + β λ i E i

6 37 G. BEN AROUS AND A. GUIONNET 6. We define another integral operator T with kernel ( T a T x y = x T y T x y + x s U y s ds + T ) y s U x s ds Then T is a symmetric Hilbert Schmidt operator in L µ WA T, since T U x s ds dµ is finite. 7. We denote by tr µ the trace in L µ WA T. 8. Let I P be the relative entropy with respect to P: I ( µ P ) log dµ = dp dµ if µ P + otherwise Proposition. (Definition). We can define a map Ɣ from into R by Ɣ µ = tr µ log ( I + β ) ( T + tr µ T exp λ T ) { } λ () exp dλ β and a map H from + WA T into R by { ( I µ P ) Ɣ µ if I ( µ P ) < H µ = + otherwise. Proof. We first show that Ɣ is well defined and finite for any µ in [see () too]. Indeed, as T is a nonnegative Hilbert Schmidt operator, tr µ log I + β T is well defined and is finite according to () for any µ + WA T. Moreover, since exp λ T is a bounded operator and T is Hilbert Schmidt for µ T exp λ T is Hilbert Schmidt and its square is trace class. Further, since T is nonnegative, tr µ T exp λ T tr µ T.So, for any µ in, the second term in the right-hand side of () exists and is bounded. Moreover, we will see later (see Lemma A.8) that, when I µ P is finite, T U x s ds dµ is finite so that µ + WA T /I µ P < +. Thus, H is well defined and finite on µ + WA T /I µ P < +. We shall prove the following theorem. Theorem.3. If β A T<, then we have the following: (i) H is a good rate function; that is, H takes its values in + and, for all M R, H M is a compact subset of + WA T. (ii) For any closed subset F of + WA T, lim sup N ( N log QN β N N δ x i i= ) F inf H F

7 SYMMETRIC LANGEVIN SPIN GLASS DYNAMICS 373 From Theorem.3, we can deduce the following quenched large deviation upper bound as in [3]. Theorem.4. almost all J, If β A T<, for any closed subset F of + WA T and for lim sup N ( N log PN β J N N δ x i i= ) F inf H F We omit the proof that Theorem.3 implies Theorem.4 since it parallels the proof given in [3], Appendix C. The strategy of the proof of Theorem.3 is the following.. First, we prove (see Section.) that Q N β is absolutely continuous with respect to P N and that the Radon Nikodym derivative of Q N β with respect to P N is equal, in the large deviation scaling, to exp NƔ ˆµ N. Hence, according to Laplace-type methods, Theorem.3(ii) is not surprising (see [] and [7]).. Once we are motivated by this last result, we study H and prove that it is a good rate function. 3. Finally, following a method very similar to the one we developed in [3], Section 3, we prove the upper bound result... Study of Q N β. We first show that QN β is absolutely continuous with respect to P N and give the Radon Nykodim derivative of Q N β with respect to P N. The Girsanov theorem implies that, for almost all couplings J, Pβ N J is absolutely continuous with respect to P N and describes its Radon Nikodym derivative. Thus, it is not hard to see that, if we denote by B i the process defined by B i t = B t x i =x i t x i + t U xi s ds, then [ dp N ] [ { M N β T = β J N T ( N ) = exp β J dp N ji x i t db j t j= N i= (3) β T ( N }] J ji xt) i dt N and we have the following proposition. Proposition.5. We have Q N β P N and dq N β dp = N MN β T In order to study the law of the empirical measure under Q N β, we want to prove that M N β T is a function of the empirical measure. More precisely, let I be the identity in the tensor product space L µ WA T L µ WA T and tr µ µ the i=

8 374 G. BEN AROUS AND A. GUIONNET trace in L µ WA T L µ WA T. We then define Ɣ µ = ( I + β 4 tr µ µ log T I + β ) I T I + β T I + β T 4 tr µ log I + β T β T tr µ I + β T T + β T 4 Denote in short ˆµ N for the empirical measure Ni= δ N x i. We are going to prove the following statement. Theorem.6. (i) We have, P N almost surely, M N β T = exp NƔ ˆµN +Ɣ ˆµ N (ii) There exists a finite constant C = C β T A such that, for any discrete probability measure on W A T, µ + WA T,ifdim µ denotes the dimension of the image of T in L µ, Ɣ µ C + dim µ / Ɣ µ + so, if D = exp C, P N almost surely, { ( D N exp N C } ( )Ɣ ˆµ N dqn β N D+ exp{ N + C } )Ɣ ˆµ N N dp N N Remark. It is obvious that T U x s ds dp x is finite. Hence, T U x s ds d ˆµ N x = /N N i= T U xi s ds is P N almost surely finite, that is, ˆµ N, P N almost surely. Thus, Ɣ ˆµ N is well defined, P N almost surely. To prove Theorem.6, we shall use spectral theory.... Spectral calculus. In the following pages, an integer N will be given. We may regard J = J ij i j N as an element of the space N of the N N real symmetric matrices. For any x x N such that T U xi s ds is finite for any i N, we define two other symmetric matrices A and B in R N N by A ij = ( T T ) x i t N dbj t + x j t db i t = ( T T ) x i T N xj T xi xj + x i s U xj s ds + x j s U xi s ds δ ijt T B ij = x i t N xj t dt Let λ i be the eigenvalues of B and e i be the eigenvectors of B in R N such that Be i = λ i e i. We prove the following.

9 SYMMETRIC LANGEVIN SPIN GLASS DYNAMICS 375 Proposition.7. We have, P N almost surely, { N M N β T = exp β e i Ae j N log + β λ + β λ i + β λ j 4 i + β λ j Proof. i j= i j= 4 If tr denotes the trace in N, since J = J,weget N T N J ji x i t dbj t = A ij J ji = tr AJ N i j= i j= N T J N ji J jk x i t xk t dt = tr JBJ =tr JBJ i j k= N i= } log + β λ i So, since denotes the expectation with respect to the Gaussian variable J, we get that, for any x = x x N such that T U xi s ds is finite for i N and so P N almost surely, M N β T = [ exp { β tr JA β tr JBJ }] Using the usual rules of computation for Gaussian variables (see [3], Proposition 8.4), we get { M N β T [exp = }] β tr JBJ (4) { [ exp β tr JA Lemma.8. [ exp { β tr JBJ }] { = exp 4 N i j= log + β λ i + λ j 4 exp / β tr JBJ exp / β tr JBJ N i= ]} } log + β λ i Proof. We have chosen the J ij i j N s so that their law is invariant by rotation on R N ; that is, for any orthogonal matrix O, the law of J ij i j N is invariant by the action J OJO. Thus, if O is an orthogonal matrix such that OBO is a diagonal matrix D = diag λ λ N, then [ exp { β tr JBJ }] = [ exp { β tr JDJ }] { } N N = exp log + β λ 4 i + λ j log + β λ 4 i i j= i=

10 376 G. BEN AROUS AND A. GUIONNET Lemma.9. [ tr JA exp / β ] tr JBJ = exp / β tr JBJ N i j= e i Ae j + β λ i + λ j Proof. Let à = OAO. Since the law of J is invariant by rotation, [ tr JA exp / β ] tr JBJ exp / β tr JBJ [ exp / β ] tr JDJ = tr Jà exp / β tr JDJ = [ exp / β ] tr JDJ à ij à kl J ji J lk exp / β ijkl tr JDJ However, [ exp / β ] tr JDJ J ij J kl exp / β tr JDJ if j i k l and l k if j i = k l or l k i j = + β λ i + λ j if i = j = k = l + β λ i + λ j Since à = Ã, we conclude [ tr JA exp / β ] tr JBJ = exp / β tr JBJ N i j= à ij + β λ i + λ j Finally, according to the definition of O, ife i is the eigenvector of B associated with the eigenvalue λ i, then Ãij = e i Ae j, so we have proved Lemma.9. According to (4), Lemmas.8 and.9 give Proposition Proof of Theorem 6. We shall now use Proposition.7 to express M N β T as a function of the empirical measure (and of N). To this end, we shall use that L ˆµ W A N T and RN are isomorphic whenever the x i s are distinct, and so P N -a.s. More precisely, we shall prove that the operator B in R N and the integral operator T on L ˆµ W A N T with kernel T x ty t dt are identical after the natural identification of R N and L ˆµ (when the x i s are distinct). For N convenience, we state without proof the following trivial identification.

11 SYMMETRIC LANGEVIN SPIN GLASS DYNAMICS 377 Proposition.. Let x x N W A T N and ˆµ N = /N N i= δ x i. (i) Let ψ L ˆµ W A N T RN Z Z x Z x N N Then ψ is an isomorphism from L ˆµ N W A T ˆµ N into RN endowed with the Euclidean scalar product. Moreover, ψ T = Bψ As a consequence, if E E are eigenvectors of T with eigenvalues λ λ, then ψ E ψ E are eigenvectors of B with eigenvalues λ λ and, for i j, ψ E i ψ E j = E i E j ˆµ N. (ii) If the x i are distinct, there exists an orthonormal basis E i i N of eigenvectors of T in L ˆµ N W A T with eigenvalues λ i i N ; T E i = λ i E i. Then ψ E i i N is an orthonormal basis of eigenvectors of B and Bψ E i = λ i ψ E i. Corollary.. Almost surely P N the operators T on L ˆµ W A N T and B on R N have the same eigenvalues and there exists a one-to-one map between their N eigenvectors. Corollary. is a direct consequence of Proposition.(ii) since, as P is the law of a diffusion, P does not put mass on points of W A T so that the xi are P N almost surely distinct. As a consequence of Proposition.7, Proposition. and Corollary., we find the following. Proposition.. We have, P N almost surely, log M N β T = 4 tr ˆµ N ˆµ log( I + β N T I + β ) I T + β N tr ˆµ N ˆµ N (( I + β I T + β T I ) T T ) β (( T tr ˆµ N I + β ) T ) T 4 tr ˆµ log( I + β ) β T N T + 4 where I denotes the identity in the tensor product space L ˆµ W A N T Lˆµ W A N T and the symmetry operator in L ˆµ W A N T Lˆµ W A N T such that, for any f g L ˆµ W A N T, f g =g f

12 378 G. BEN AROUS AND A. GUIONNET Proof. We stated in Proposition.7 that (5) log M N β T = 4 N i j= + β N i j= log + β λ i + β λ j 4 e i Ae j + β λ i + λ j N log + β λ i i= According to Corollary., P N -a.s, the operators B in R N and T in L ˆµ N W A T have the same eigenvalues λ i i N, so that (6) and (7) N log + β λ i =tr ˆµ N log I + β T i= N log + β λ i + β λ j =tr ˆµ N ˆµ log I + N β T I + β I T i j= We now focus on N i j= e i Ae j / + β λ i + λ j. It is an easy matter to see that this term does not depend on the choice of the basis of eigenvectors of B. Let E i i N be an orthonormal basis of eigenvectors of T. We choose e i = ψ E i i N as in Proposition.(ii). Then e i Ae j = N N k l= E i x k A k l E j x l N = N 3/ E i x k ( T x k t db t x l + k l= = N / E i T E j ˆµ N T N / δ ij T ) x l t db t x k E j x l so that (8) However, N i j= e i Ae j N + β λ i + λ j = N i j= N T i= E i T E j ˆµ N + β λ i + λ j E i T E i ˆµ N + β λ i + T 4 E i T E j ˆµ = E N i E j T T E i E j ˆµ N ˆµ N + β λ i + λ j = E i E j I + β I T + β T I E i E j ˆµ N ˆµ N

13 SYMMETRIC LANGEVIN SPIN GLASS DYNAMICS 379 and since E i E j i j N is an orthonormal basis of the tensor product space L ˆµ W A N T Lˆµ W A N T, we deduce that (9) N E i T E j ˆµ N i j= + β λ i + λ j = tr ˆµ N ˆµ N (( I + β I T + β T I ) T T ) Equations (5) (9) achieve the proof of Proposition.. Proof of Theorem.6(i). We show here that Theorem.6(i) is equivalent to Proposition.. In fact, we can see that (( β tr ˆµ N ˆµ I + β I N T + β T I ) T T ) = tr ˆµ N T exp λ T exp { λβ } dλ in view of the following resolvent formula. Lemma.3. β ( I + β I T + β T I ) = exp λ T exp λ T exp { λβ } dλ The proof of this lemma is trivial as soon as we notice that this equality is true on the orthonormal basis E i E j i j N of L ˆµ W A N T Lˆµ W A N T. Thus, by definition of Ɣ, Proposition. implies that log M N β T = N tr ˆµ log( I + β ) N T + Ɣ ˆµ N + N tr ˆµ N T exp λ T exp { λβ } dλ = NƔ ˆµ N +Ɣ ˆµ N Proof of Theorem.6(ii). We finally bound Ɣ. Lemma.4. There exists a finite constant C = C β A T such that, for any probability measure µ on W A T, Ɣ µ C dim µ / + + Ɣ µ where dim µ denotes the dimension of the image of T in L µ.

14 38 G. BEN AROUS AND A. GUIONNET () Proof. Let λ i be the eigenvalues of T in L µ WA T. Then Ɣ µ = ( + β λ i + β ) λ j log 4 + β λ i + β λ j 4 i j= log + β λ i + β T Tβ 4 i= i= E i T E i µ + β λ i Since the λ i s are nonnegative, for any i N, log + β λ i β λ i.so () log + β λ i β λ i = β tr µ T β A T i= i= Moreover, for any positive real numbers a b, we have the elementary inequality So () exp ab + a + b + a + b β A T β 4 tr µ T = β 4 i j= i j= λ i λ j ( + β ) λ i + λ j log + β λ i + β λ j Finally, we observe that for any real numbers a b and any positive α, we have ab αa + α b so that (3) E i T E i µ + β i λ dim µ / i i E i T E i µ + β λ i + dim µ / dim µ / Ɣ µ +β A T +dim µ / where we have used the bound () in the last line. Lemma.4 is a direct consequence of () (3)..3. Continuity properties of Ɣ. In order to study the rate function H and to prove the large deviations upper bound theorem, we first have to study the map Ɣ. Since this study is rather heavy and technical, we will only state the results here, leaving the proofs and details in the Appendix. To this end, let us first define linear functions ν which are given, for any probability measure ν on W A T,by T T ν µ = β ds dt I + β T X s X t ν U x s U x t µ

15 SYMMETRIC LANGEVIN SPIN GLASS DYNAMICS 38 Then Ɣ can be approximated by the sum of a continuous function and a linear function in the following sense. Proposition.5. There exists a finite constant C such that, for any probability measure µ in, for any probability measure ν in + WA T, for any positive real number M, there exists a bounded continuous function Ɣ M such that Ɣ µ Ɣ M µ ν µ ( ) C M + d T µ ν C µ where d T is the Wasserstein distance which is defined by { (4) d T µ ν =inf sup x s x s dξ x x s T where the infimum is taken on the probability measures ξ on W A T WA T with marginals µ and ν and C µ = T x t dt dµ x 3/ + s..4. H is a good rate function. Let us now show that H is a good rate function, that is, Theorem.3(i). We first prove that H is nonnegative. This fact is not trivial since we cannot prove a large deviation lower bound. In order to see that, we first derive an alternative expression for Ɣ, which will also be useful for identifying the minima of H..4.. An alternative expression for Ɣ. We denote by X s the evaluation at time s, that is, the map from W A T into R such that for any x WA T, X s x =x s. We denote by a t the function in L µ WA T L µ WA T defined by ( t t ) a t = X t X t X X + X s U X s ds + U X s X s ds According to Itô s formula, a t is also given, under any probability measure µ P, by ( t t a t = db s X s + X s db s ) o where B t x =x t x + t U x s ds We then define, for any probability measure µ in, a function F µ in L µ WA T by F µ ( t x = y t I + β t I + β ) at I t x y dµ y Let µ satisfy I P <. Then B is a semimartingale under µ according to Girsanov s theorem. Moreover, F µ is previsible and belongs to L µ WA T,so that T Fµ s db s is well defined under µ and belongs to L µ WA T. Lemma.6. Let µ I P < + ; then ( T Ɣ µ = β F µ t x db t x β4 T } / ) F µ t x dt dµ x

16 38 G. BEN AROUS AND A. GUIONNET Proof. Let T = I + β T I + β I T. By the definition of Ɣ,wehave Since a T is the kernel of T,weget β Ɣ µ =tr µ µ T T T β Ɣ µ = a T T a T µ = T T X 4 t db t + db t X t T ( T T ) X t db t + db t X t where we write µ instead of µ µ for the scalar product in L µ WA T L µ WA T for simplification. At this point, we have not used the existence of stochastic integrals against B since a T is pointwise defined. We shall now take into account that we suppose that I µ P is finite, so that µ P and T X t db t is well defined in L µ WA T L µ WA T. Since T is symmetric, we get (5) β Ɣ µ = T X t db t T ( T T ) X t db t + db t X t µ We want to apply Itô s formula in (5). To this end, we study the martingale properties of the processes contained in the bracket of the right-hand side of (5). We first observe that µ (6) T T T X t db t = I + β T I X t db t T β I + β T I T I T X t db t However, T I T X t db t x y = dµ z T y t z t dt T x t db t z so that, using the semimartingale representation of B, we see that I T T X t db t has finite variations. As a consequence, I+β T I T I T T X t db t has finite variations. Moreover, I+β T I T X t db t = T I+β T X t db t and, for any y W A T, s I+β T X t y db t s T is a martingale under P with martingale bracket with s y t db t equal to s y t I + β T X t y dt.

17 SYMMETRIC LANGEVIN SPIN GLASS DYNAMICS 383 As a conclusion, (6) implies that, for any y W A T, s T X t db t y s T is a semimartingale whose martingale bracket with s y t db t is equal to s y t I + β T X t y dt. Hence, Itô s formula implies that T T X t db t T X t db t µ ( T T ) = y t db t x T X t db t y x dµ x dµ y T t T = X t db t T X s db s + X t I + β T X t µ dt Similarly, we find T T X t db t T db t X t so that we have proved T (7) β Ɣ µ = db t X t T a t + µ µ µ T t = X t db t T db s X s µ T X t I + β T X t µ dt We now focus on the dependence of T on the time variable T. Let s be an integral operator in L µ WA T with kernel d s x y =x s y s. Then we state the following. Lemma.7. For any probability measure µ in W A T, for any f g in L µ WA T L µ WA T, and T f T g µ = f g µ β f t I t + t I t g µ dt Proof. Let n = = t <t < <t n+ = T be a subdivision of T. Let n =max k n t k+ t k and let Then m = m tk t k t k = k= m = ( I + β m I + β I m ) (8) n β k= k I tk + tk I k+ t k+ t k =I n+

18 384 G. BEN AROUS AND A. GUIONNET To prove Lemma.7, we can assume without loss of generality that f = g = f f where f f are in L µ WA T and satisfy f dµ = f dµ =. Then f f T I + I T n+ I I n+ f f µ n tk+ ( = f X t X tk µ + f X t X tk ) µ dt t k k= n tk+ k= t k x t x tk dt dµ x But the canonical process is bounded and continuous under µ, so that lim n n tk+ k= t k x s x tk ds dµ x = Hence lim f f T I + I T n+ I I n+ f f µ = n Since T and n+ are positive operators, we deduce lim f f n+ T f f µ = and, similarly, lim n n f f = T n k I tk + tk I k+ t k+ t k f f µ k= f f t I t + t I t f f µ dt So (8) gives Lemma.7 when n tends to zero. Since t X s db s and t db s X s (and so a t ) belong to L µ WA T L µ WA T for any t T, we can apply Lemma.7 in (7). We find T β Ɣ µ = db t X t t a t + T X µ t I + β T X t µ dt T (9) β t I t + t I t ( t t s db s X s + X s db s ) db u X u dt µ

19 SYMMETRIC LANGEVIN SPIN GLASS DYNAMICS 385 We can use Itô s formula for the last term of the right-hand side of (9) as we did previously to prove (7). We find ( t t s t I t + t I t db s X s + X s db s ) db u X u µ t () = I t + t I t a t t db u X u t t I + β t X s I + β t X s µ ds Moreover, since the kernel of t is d t x y =x t y t,wefind µ and () t I + β t X s I + β t X s µ = X t I + β t X s µ I t + t I t a t t t = dµ x + = dµ x db u X u µ t dµ y y t t db u X u x y dµ z z t t a t x z dµ y y t t t dµ x ( dµ z z t t a t z x ) db u X u y x dµ z z t t a t z x where () comes from the symmetry of the function x z t a t x z. Let F µ t x = dµ y y t t a t x y Then () reads t I t + t I t a t t Thus, () becomes ( t t I t + t I t = and so (9) shows Ɣ µ = () db u X u db s X s + t = µ X s db s ) F µ t x dµ x X t I + β t X s µ ( dµ x + β + β4 T T T β F µ t x db t x β4 X t I + β T X t µ dt dt t F µ t x dµ x T X t I + β t X s µ ds s db u X u ) F µ t x dt µ

20 386 G. BEN AROUS AND A. GUIONNET We can compute (3) T T β X t I + β T X t µ dt + β 4 dt t X t I + β t X s µ ds T = β X t I + β t X t µ dt = tr µ log I + β T Equations () and (3) complete the proof of Lemma H is nonnegative. Lemma.8. The rate function H maps + WA T into R+ ; that is, for any µ satisfying I P < +, (4) Ɣ µ I µ P Proof. Let µ I P < +. We can apply Lemma.6. Since F µ t x is a previsible function along the canonical filtration t = σ x s s t, under P, M µ t x = β t Fµ s x db s x is a local continuous martingale along the filtration t t T, with quadratic variation M µ t = β 4 t Fµ s x ds. Let τ K = inf t/ M µ t Mµ t > K. Since M µ is continuous, τ K is a stopping time for the canonical filtration. As a consequence, m µ T τ K = M µ T τ K < Mµ > T τk is measurable. According to the definition of τ K, m µ T τ K is bounded by K. We now apply the relative entropy property, (5) m µ T τ K x dµ x I µ P +log exp m µ T τ K x dp x But exp m µ t τ K t T is a bounded martingale with respect to the filtration t τk t T. Hence, for any positive real number K, exp m µ T τ K x dp x = exp m µ x dp x = so that (5) becomes (6) m µ T τ K x dµ x I µ P Thus, to deduce Lemma.8 from (6), we need to show that: (7) m µ T τ K x dµ x = m µ T x dµ x lim K Since m µ T τ K m µ T and mµ T τ K x converges to m µ T x when K tends to infinity for any x such that m µ T x is finite, the dominated convergence theorem shows that (7) is satisfied as soon as m µ T belongs to L µ. To establish

21 SYMMETRIC LANGEVIN SPIN GLASS DYNAMICS 387 this last point, we only need to prove that M µ T = T Fµ t x dt belongs to L µ. However, since T is positive, T F µ t x dt dµ x T 4A t a t x y dµ x y dt T ( t t A y s db s x + x s db s y ) dµ x y dt Using the relative entropy property and the monotone convergence theorem, we conclude that [see (9)], for any positive real number α small enough, there exists a finite constant ξ such that T F µ t x dt dµ x α A TI µ P +ξ Thus, for any µ in I P < +, m µ T belongs to L µ so that (6) and (7) imply that is, Corollary.8. m µ T x dµ x I µ P.4.3. H is a good rate function. We first show that the entropy relative to P is bounded in terms of H. Lemma.9. If β A T<, there exists a strictly positive real number α and a finite constant C, C>, such that H µ αi µ P C Proof. Let µ + WA T. If I µ P =+, then H µ =+ so that Lemma.9 is true. Otherwise, I µ P is finite so that H µ =I µ P Ɣ µ. Moreover, Ɣ µ =Ɣ µ +Ɣ µ Ɣ µ but Ɣ µ =β ( tr µ µ I + β t I + β I t T T ) β tr µ T = β 4 ( T x t db t y + T y t db t x ) dµ x y

22 388 G. BEN AROUS AND A. GUIONNET Thanks to the relative entropy properties, for any x W A T and any positive real number κ: ( κ β T T x 4 t db t y + y t db t x ) dµ y ( I µ P +log exp {κ β T T ) } x 4 t db t y + y t db t x dp y and then, for any positive real number ε, (8) ( κε β T T x 4 t db t y + y t db t x ) dµ x y + ε I µ P ( + log exp {κε β T T ) } x 4 t db t y + y t db t x dp x y Let J be a centered Gaussian variable with covariance. ( exp {κε β T T ) } x 4 t db t y + y t db t x dp x y [ { ( κε T T )} ] = exp βj x t db t y + y t db t x dp x y [( { ( T T )} ) / ] (9) exp κεβ J x t dt + y t dt dp x y [ { T = exp κεβ J exp κεβ A TJ = } ] x t dt dp x κεβ A T where the last equality holds as soon as κεβ A T<. However, we supposed that β A T< in order to choose κε > small enough so that κεβ A T<. We then choose ε> so that + ε<κε. Hence, inequalities (8) and (9) show that we can find a strictly positive real number α = κε ε /κε and a finite constant C = / κε κεβ A T such that so that Ɣ µ α I µ P +C H µ αi µ P C We now prove that H is lower semicontinuous. We assume in the following that β A T<. Take a sequence µ k of probability measures converging to a probability measure µ and choose a subsequence n k such that lim inf k H µ k =lim k H µ nk.

23 SYMMETRIC LANGEVIN SPIN GLASS DYNAMICS 389 We distinguish the case where I µ nk P stay bounded for large k from the case where we can find a subsequence µ nk K such that I µ nk K P K. In the first case, we suppose that I µ nk P stay bounded for k larger than some k. Then C µ is uniformly bounded by a finite constant L for k k, according to Lemma A.8. Moreover, Lemma.5 says that, for any positive real number M, for any k k, Ɣ µ nk Ɣ M µ nk µ µ nk ( ) (3) C L M + d T µ µ nk Hence, for any k k, ( ) (3) H µ nk I µ nk P Ɣ M µ nk µ µ nk C L M + d T µ µ nk Let Q µ be a probability measure on W A T, absolutely continuous with respect to P, such that dq µ dp x = exp Z µ where { T Z µ = exp β { β T T T } I + β T x s x t µ U x s U x t dt ds } I + β T x s x t µ U x s U x t dt ds dp x In the regime β A T<, Z µ is finite. Note that dq µ dp x = Z µ exp µ δ x Then, we can prove as in Appendix B of [3] that (3) I P µ = I Q µ log Z µ so that (3) becomes ( ) (33) H µ nk I µ nk Q µ log Z µ Ɣ M µ nk C L M + d T µ µ nk Since I Q µ is l.s.c and Ɣ M is continuous, (33) gives lim inf n H µ n = lim k H µ nk I µ Q µ log Z µ Ɣ M µ C L M = I µ P µ µ Ɣ M µ C L M by (3) H µ C L M by (3) Since the last inequality holds for any real number M, we conclude that lim inf n H µ n H µ.

24 39 G. BEN AROUS AND A. GUIONNET In the other case, we can find a subsequence n p K such that lim K I µ np K P =+ and then Lemma.9 implies that lim inf k H µ k = lim k H µ nk = lim K H µ n p K =+ so that we also get lim inf k H µ k H µ Moreover, H is a good rate function. Indeed, for any positive real number R, H R is a compact set as it is a closed set (H is l.s.c) which is included in a compact set, since, by Lemma.9, the relative entropy I P is bounded on H R..5. Proof of the large deviation upper bound. As in the asymmetric version of dynamics, we first prove an exponential tightness result, and we then prove a weak large deviation upper bound, that is, Theorem.3(ii) when F is compact. We finally deduce from these two results Theorem.3(ii) for any closed subset F. Lemma.. such that If β A T<, there exists α> and a finite constant C ( dq N ) α β dp N C N dp N Proof. With the notation of Section.., ( dq N ) α [ { β dp N = exp βtr JA }] α dp N β Tr JBJ dp N [ { exp αβtr JA }] β αtr JBJ dp N Let p q be conjugate exponents. The Hölder inequality gives ( dq N ) α [ β dp N dp exp{ N αpβtr JA }] /p dp N β α p Tr JBJ [ { }] /q dp N exp qβ α pα Tr JBJ Recall that { exp αpβtr JA } β α p Tr JBJ { = exp αpβ N i= T β α p N ( N ) J ij x j t db j t N i= T j= ( N N j= ) } J ij x j t dt

25 SYMMETRIC LANGEVIN SPIN GLASS DYNAMICS 39 is a supermartingale, so that we find, for conjugate exponents p q, (34) ( dq N ) α [ β dp N dp N exp ] /q dp N qβ α pα Tr JBJ But, if λ i i N are the eigenvalues of B, we can prove as in Lemma.8 that [ exp qβ α pα Tr JBJ ] (35) ( = exp N 4 i j= log qβ α pα λ i + λ j N 4 i= ) log qβ α pα λ i whenever α is close enough to one. Indeed, since the λ i s are positive, N λ i λ i = N N i= i= T x i t dt A T P N -a.s. But we supposed that β A T<, so we can find α> small enough and two conjugate exponents p and q such that max i j ( qβ α pα λ i + λ j ) qβ α pα A T<. Then, the right-hand side of (35) is finite. More precisely, we can find a finite constant c such that, for any x smaller than qβ α pα A T< (see Appendix B of [3]), log x cx so that equality (35) implies [ exp qβ α pα Tr JBJ ] exp cqβ α pα A T ( exp { cqβ α pα A T }) N which proves Lemma.. We turn to the proof of the weak upper bound. Lemma.. If β A T<, for any compact subset K of + WA T, lim sup N N log QN β ˆµN K inf K H Proof. Let K be a compact subset of + WA T. For any positive real number δ, we can cover K by a finite number p of open balls B µ i δ for Wasserstein s distance d T : B µ i δ = ν + WA T /d T µ i ν <δ K B µ i δ i p

26 39 G. BEN AROUS AND A. GUIONNET Let L be a positive real number and let L be defined by { / ( T } L = µ + WA T U x t dt) dµ x L We then bound Q N β ˆµN K : (36) Q N β ˆµN K Q N β ˆµN c L + p Q N β i= ( ˆµ N K L B µ i δ ) (a) Estimate of Q N β ˆµN L c. We use the Hölder inequality with the real number α> introduced in Lemma. and its conjugate exponent σ: (37) Q N β ˆµN c L = ( ˆµ N c L ˆµ N c L dq N β dp N dp N ( dq N β dp N C N P N ˆµ N c L /σ ) α dp N ) /α P N ˆµ N c L /σ Using Chebyshev s inequality, we get, for any positive real number r, [ ( N T ) ] P N ˆµ N L c exp rnl exp r U x i t dt dp N (38) i= [ ( T ) ] N exp rnl ( exp r U x t dt dp) But, if r is small enough, exp r T U x t dt dp is finite (see the proof of Lemma A.8), so that, in conclusion of (37) and (38), we find, in the high temperature regime β A T <, a strictly positive real number r and a finite constant D such that we can state the following. Lemma.. For any positive real number L, Q N β ˆµN L c exp r L D N (b) Estimate of Q N β ˆµN L K B µ i δ. According to Theorem.6, Q N β ˆµN L K B µ i δ ( D + N exp N + C )Ɣ ˆµ N dp N N L K B µ i δ so that the Hölder inequality and Lemma. imply lim sup N N log QN β ˆµN L K B µ i δ lim sup exp NƔ ˆµ N dp N N L K B µ i δ

27 SYMMETRIC LANGEVIN SPIN GLASS DYNAMICS 393 However, we saw in Proposition.5 that, for any probability measure µ in L, ( ) Ɣ µ Ɣ M µ µi µ C M + d T µ µ i C µ On the subset L K B µ i δ, C µ is uniformly bounded by m L = L 3/ + and d T µ i µ by δ, so that lim sup exp NƔ ˆµ N dp N N L K B µ i δ (39) ( C )m M + δ L + lim sup exp N Ɣ M ˆµ N + µi ˆµ N dp N N K B µ i δ Let Q µi be a probability measure on W A T, absolutely continuous with respect to P such that dq µi dp = exp Z µi δ x µi The measure Q µi is well defined in the regime β A T< and Z µi is then finite. Then (38) reads (4) Q N β ˆµN L K B µ i δ ( ) } M CZ N µ i exp {C + δ m L N Using Sanov s theorem, we deduce (4) lim N exp N Ɣ M ˆµ N d Q µi N K B µ i δ N log QN β ˆµN L K B µ i δ ( log Z µi + C )m M + δ L inf I Q µi Ɣ M K B µ i δ As in Appendix B of [3], we find that { I µ P µi µ +log Z µi if I µ P < + I µ Q µi = + otherwise. However, we recall (see Lemma A.5) that ( T µi µ µ c d T µ µ i U x t dt) dµ

28 394 G. BEN AROUS AND A. GUIONNET and since we saw in the proof of Lemma A.8 that there exist two finite constants c and c such that ( T U x t dt) dµ c I µ P +c we find two finite constants C and C such that (4) becomes (4) lim sup N N log QN β ˆµN L K B µ i δ ( C )m M + δ L inf C δ I µ P µ Ɣ M µ µ K B µ i δ If we recall (36), Lemma. and (4), we proved that (43) lim sup N N log QN β ˆµN K { max r L D ( } C )m M + δ L inf C δ I µ P µ Ɣ M µ µ K We now need to show the following. Corollary.3. ( lim lim inf C δ I µ P µ µ Ɣ M µ ) inf H δ M µ K K To this end, we give a technical lemma. Lemma.4. If β A T<, there exists α<and a finite constant ξ such that, for any positive real number M, for any probability measure µ, Ɣ M µ + µ αi µ P +ξ The proof of Lemma.4 follows the lines of the proof of Lemma.9; we omit it. Proof of Corollary.3. We choose δ small enough such that κ = C δ α>, so that Lemma.4 implies (44) C δ I µ P µ Ɣ M µ κi µ P ξ so that, if we distinguish the case where inf K H = inf K I µ P Ɣ µ is finite from the case where it is not, we find as follows: (i) if inf K H =+, then inf K I µ P =+ so that (44) implies that, for any positive real number M, inf µ K C δ I µ P µ µ Ɣ M µ =+ ;

29 SYMMETRIC LANGEVIN SPIN GLASS DYNAMICS 395 (ii) if inf K H<+, since H is a good rate function, we can find a finite real number R such that inf H = inf H K K I R However, for any real number R, (44) implies inf C δ I µ P µ µ Ɣ M µ µ K (45) { } max inf C δ I µ P µ Ɣ M µ κr + ξ µ K I R By Lemma.5, we know that, for any probability measure µ in, Ɣ µ Ɣ M µ µ C M C µ so that Lemma A.8 shows that there exists a finite constant k such that, for any real number R, sup Ɣ Ɣ M k R I P R M( 3/ + ) Thus inf C δ I P Ɣ M K I R inf I P Ɣ C δr k K I R M R 3/ + = inf H C δr k K I R M R 3/ + Therefore, (45) implies that, if c = max k C, ( inf C δ I µ P µ Ɣ M µ ) µ K { max inf K I R ( H + c δ + M) ( R 3/ + ) } κr + ξ so that, for any real number R R, lim lim inf C δ M δ I P Ɣ M K { } { } max inf H κr + ξ = max inf K I R K H κr + ξ We finally let R + so that we get Corollary.3. We can end the proof of Lemma.. If we let δ tend to zero and M tend to infinity in (43), Corollary.3 implies that, for any positive real number L: { } lim sup N N log QN β ˆµN K max r L D inf H K So that, letting L tend to infinity proves Lemma..

30 396 G. BEN AROUS AND A. GUIONNET We finally deduce Theorem.3(ii) from Lemmas. and.; that is, we show the following. Lemma.5. If β A T<, for any closed set F of + WA T, lim sup N N log QN β ˆµN F inf H F Proof. It is well known (see, for instance, [8], Lemma 3..7) that the law of the empirical measure under P N is exponentially tight; that is, for any real number L, there exists a compact subset K L of + WA T so that P N( ˆµ N K c L) exp LN Then Lemma. implies that the law of the empirical measure under Q N β is exponentially tight since, if α> is chosen as in Lemma. and σ is the conjugate exponent of α, wehave ( ( dq N ) α ) /α Q N β ˆµN K c L β dp N P N ˆµ N K c dp N L /σ C /α N exp { Lσ } N Thus, the weak large deviation upper bound of Lemma. implies Lemma.5 (see [8], Lemma.5, page 4). 3. Existence and uniqueness of the minima of the rate function. We shall use Theorem.3 to study the convergence of the law N β T of the empirical measure under Q N β. We recall that, for any probability measure µ in, we defined in Lemma.6 a function F µ t on W A T by F µ ( t x = dµ y y t I + β t I + β ) I t at x y Then we have the theorem. Theorem 3.. The rate function H achieves its minimum value = at a unique probability measure Q on W A T which is implicitly defined by Q P dq { T x =exp β F Q t x db dp t x β4 T } F Q t x dt We can also give a pathwise description of the minima of H. Corollary 3.. The good rate function H achieves its minimum value at a unique probability measure Q which is the solution of the nonlinear stochastic differential equation dx t = U x t dt + db t + β F Q t x dt Law of x = Q Law of x = Q = µ

31 SYMMETRIC LANGEVIN SPIN GLASS DYNAMICS 397 The proof is a direct consequence of Girsanov s Theorem which implies that Theorem 3. and Corollary 3. are equivalent (see [3], Theorem 6.3, for more details). Moreover, when H is l.s.c., we know that H achieves its minimum value. As a consequence, if β A T<, there exists a unique solution of the nonlinear stochastic differential equation described above. Furthermore, Q N β being an exchangeable law, a result due to Sznitman (see Lemma 3. in [5]) allows deducing from Theorem 3. the propagation of chaos result. Corollary 3.3. Let β A T<. (i) N β T converges weakly to δ Q. In particular, if F is a bounded continuous function on + WA T, then F ˆµ N dq N β = F Q lim N (ii) For any bounded continuous functions f f m on W A T, lim f x f m x m dpβ N J x dγ = m f i x dq N We can also deduce (as in [3], Appendix C) from Theorems.4 and 3. that the quenched law of the empirical measure converges exponentially fast to δ Q, so we have the following. Corollary 3.4. all J, Let β A T<. (i) If F is a bounded continuous function on + WA T, then, for almost lim F ˆµ N dpβ N J =F Q N (ii) For almost all J and for any bounded continuous function f on W A T, lim N N N f x i = fdq i= The proof of Theorem 3. will need two steps. First, we shall prove that H achieves its minimum value on the set M of probability measures on W A T defined by { M = Q/Q P dq { T x =exp β F Q t x db dp t x β4 T ( Q F t x ) dt }} In a second step, we shall prove that M is reduced to a unique probability measure. a.s. i=

32 398 G. BEN AROUS AND A. GUIONNET 3.. Study of the minima of H. We first prove that any minimum of H is equivalent to P. Lemma 3.5. If Q minimizes H, then Q is equivalent to P. Lemma 3.5 is a straightforward consequence of Lemma 3.6. Lemma 3.6. Let Q be a probability measure on W A T which minimizes H. Then we have the following conditions: (i) Q P; (ii) denote B = x W A T / dq/dp x = and δ = P B, (a) I Q + s B P/ + sδ P =I Q P +sδ log s + O s (b) if β A T<, Ɣ Q + s B P / + sδ =Ɣ Q +O s, so that ( ) Q + s B P H H Q =δs log s + O s + sδ Remark 3.7. We do not think that the condition β A T < is really crucial in Lemma 3.6(ii)(b) but we leave it since we are not able to prove any large deviation upper bound result without it. Proof. (i) Since I Q P is finite, Q P. (ii)(a) One can compute ( ) Q + (46) s B P I P = + sδ + sδ I Q log + sδ P + sδ + sδ + sδ log s + sδ which gives (ii)(a). (ii)(b) We state a result even stronger than Lemma 3.6(ii)(b). Lemma 3.8. If β A T<, for any probability measure µ in, and for any signed measure ν such that ν W A T = and T U x s ds d ν is finite and for which µ + δν is a probability measure when δ is small enough, Ɣ is Gateaux-differentiable at µ in the direction ν. This lemma can be proved by expanding Ɣ and Ɣ in powers of β (which can be done under the assumption that β A T<) and then by showing that each term of these expansions are Gateaux-differentiable in a neighborhood µ + κν κ δ of µ and that the series of these derivatives is absolutely and uniformly bounded on this neighborhood. We leave the proof to the reader. We now prove that, if Q minimizes H, then Q belongs to M. Lemma 3.9. If Q minimizes H, then Q is the solution of the nonlinear equation Q P dq { T dp = exp β F Q t db t β4 T } F Q t dt

33 SYMMETRIC LANGEVIN SPIN GLASS DYNAMICS 399 To prove Lemma 3.9, we study the Taylor expansion of H at Q in the direction of ν = ψ Q, for bounded measurable functions ψ such that ψdq=. Lemma 3.. Let β A T<. (i) I + sψ Q P =I Q P +s log dq/dp ψdq+ o s. (ii) Ɣ + sψ Q = Ɣ Q +s β T FQ t db t β 4 / T FQ t dt + Y T ψdq+ o s, where Y s s T is the previsible process with finite variations defined by { s s } Y s y = β h Q t x y db t x β 4 F Q t x h Q t x y dt dq x if h Q t x y =DF Q t δ y x. The reader can prove Lemma 3. using Lemma.6. Proof of Lemma 3.9. Since Q minimizes H, lim H + sψ Q H Q = s Hence, according to Lemma 3., { log dq T dp β F Q t db t + β4 T F Q t dt Y T }ψdq= Since this equality is true for any bounded measurable function ψ such that ψdq=, we deduce that there exists a finite constant cq such that, Q almost surely, and so P almost surely by Lemma 3.5, log dq T dp = β F Q t db t β4 T F Q t dt + Y T + c Q However, dq/dp t t T must be a local martingale (see [4], Chapter VIII) so that, by uniqueness of the semimartingale decomposition, log dq T dp = β F Q t db t β4 T F Q t dt s 3.. Existence and uniqueness problem for the minima of H. The aim of this section is to prove that M is reduced to a unique probability measure Q, that is, that the rate function H achieves its minimum value at a unique probability measure Q. We will first show that H achieves its minimum value at a unique probability measure. Independently, we can construct this minimum in the regime 3β A T<. Theorem 3.. (i) For any time and temperature, there exists at most one probability measure Q such that I Q P < + which is a solution of Q P dq { T dp = exp β F Q t x db t x β4 T } F Q t x dt

34 4 G. BEN AROUS AND A. GUIONNET (ii) If 3β A T<, there exists a unique probability measure Q such that I Q P < + which is a solution of Q P dq { dp = exp β T F Q t x db t x β4 T } F Q t x dt We shall use a fixed point argument to prove Theorem 3.. To this end, we first study the functions F µ, and, more precisely, show the following. Lemma 3.. s T, For any probability measure µ in I P < + and for any { s s }] E P [exp β F µ t dt F µ t db t β4 so that exp β s Fµ t x db t x β 4 / s Fµ t x dt s T is a P t martingale. To prove Lemma 3., we show the following. Lemma 3.3. For any probability measure µ such that I µ P <, there exists a bounded previsible process f µ such that F µ t x = t I + β t X t X s µ db s + f µ t x and there exists a finite constant c such that, for any µ such that I µ P <, sup x W A T sup f µ t x c + I µ P t T Proof. Denote V t = U X t and recall that ( t t F µ t x = dµ y y t t X s db s + db s X s ) y x where, according to Lemma.3, t = dλ exp { λβ } (47) exp λ β t exp λ t Let µ I P < +. Then B is a semimartingale under µ so that we can write t F µ t x = dµ y y t t X u db u x y (48) t + dµ y y t t X u db u y x