ByG.BenArousandA.Guionnet Ecole Normale Superieure And . - Free Download PDF

Today
3 Views
0 Downloads
316.23 KB
56 Pages
Transcription

The Annals of Probability1997, Vol. 25, No. 3, 1367–1422SYMMETRIC LANGEVIN SPIN GLASS DYNAMICSBy G. Ben Arous and A. GuionnetEcole Normale Superieure and Université de Paris SudWe study the asymptotic behavior of symmetric spin glass dynamics inthe Sherrington–Kirkpatrick model as proposed by Sompolinsky–Zippelius.We prove that the averaged law of the empirical measure on the path spaceof these dynamics satisfies a large deviation upper bound in the high temperature regime. We study the rate function which governs this largedeviation upper bound and prove that it achieves its minimum value ata unique probability measure Q which is not Markovian. We deduce anaveraged and a quenched law of large numbers. We then study the evolution of the Gibbs measure of a spin glass under Sompolinsky–Zippeliusdynamics. We also prove a large deviation upper bound for the law of theempirical measure and describe the asymptotic behavior of a spin on pathspace under this dynamic in the high temperature regime.1. Introduction. The Sherrington–Kirkpatrick (S–K) model is a meanfield simplification of the spin glass model of Edwards–Anderson. The behaviorof its static characteristics, such as its partition function, has been intensivelystudied by physicists (see [12] for a broad survey). There are few mathematicalresults available (except for [1], [6], [9] and [17]).In [12], it is argued that studying dynamics might be simpler since it avoidsusing the “replica trick” and the Parisi ansatz for symmetry breaking, whichare 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 andZippelius (see [15]), who chose a Langevin dynamics scheme.In [3], we followed this strategy for asymmetric dynamics (which are notdirectly relevant to the study of statics for the S–K model). We obtained therea full large deviation principle for path space empirical measure averaged onthe Gaussian couplings (for short times or large temperatures). This largedeviation principle enabled us to derive the so-called self-consistent limitingdynamics, 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 crypticform.We can do this only in a short time or high temperature regime, and so thisprevents us from drawing any conclusion for the behavior in large time, atfixed temperature, which would be a line of attack to study the equilibriumReceived September 1995; revised October 1996.AMS 1991 subject classifications. 60F10, 60H10, 60K35, 82C44, 82C31, 82C22.Key words and phrases. Large deviations, interacting random processes, statistical mechanics,Langevin dynamics.1367

1368G. BEN AROUS AND A. GUIONNETmeasure. Weaker results concerning these dynamics are proved in [11] for anytime and temperature.To be more specific, let us recall that the S–K Hamiltonian is given, forx x1 xN 1 1 N , byN 1 NHJ x Jij xi xj N i j 1where the randomness in the spin glass is here modeled by the Jij i j whichare independent centered Gaussian random variables, and where Jij Jji .The Gibbs probability measure one would like to study (for N large) is givenbyNexp βHJ x Nα dx ZN J where α 12 δ 1 δ1 and β is the inverse of temperature.Here ZN J is the partition function 1NZN J Nexp βHJ x 2 x 1 1 NIf one replaces the hard spins 1 1 by continuous spins, that is, by spinstaking values in R, or as we shall see in a bounded interval of R, and if onereplaces the measure α 12 δ 1 δ1 by α dx e 2U x / e 2U x dx dx,where U is, for instance, a double well potential on R, then the Langevindynamics for this problem are given by βjjjdxt dBt U xt dt (1)Jji xit dt N 1 i Nwhere B is an N-dimensional Brownian motion.We want to understand the limiting behavior (for large N) of the law, sayNPβ J , of these randomly interacting diffusions given the initial law, say µ N0 .As in [3], we will only study bounded spins; that is, we will assume thatU x is defined on a bounded interval A A and tends to infinity when x A sufficiently fast to insure our spins xj stay in the interval A A .However, we will not assume as in [3] that the the whole matrix Jij i jis made of i.i.d N 0 1 random variables but rather assume the symmetryof couplings; that is, we will here suppose that the random matrix Jij i j issymmetric, that is, Jij Jji . More precisely, we will suppose that under thediagonal, the Jij ’s are i.i.d N 0 1 and N 0 2 on the diagonal. Such a choiceof covariance is nice from the technical point of view since it makes the law ofthe Jij ’s invariant by rotation. On the other hand, it does not interfere withthe limit behavior of the spin glass.So, under this symmetry hypothesis, our dynamics (1) are reversible andtheir invariant measure is given by the Gibbs measure: NN NNiµJ dx exp βHJ x 2U x dxi i 1i 1

SYMMETRIC LANGEVIN SPIN GLASS DYNAMICS1369Thus the symmetry hypothesis is crucial to understanding S–K dynamics. Onthe other hand, this model is much more difficult to understand than theasymmetric one. Our first goal is to study the empirical measure µ̂N 1/N Ni 1 δxi onpath space. There is no reason for this to be a simple problem, since, forfixed interaction J, the variables x1 xN are not exchangeable. So wefirst study the law of the empirical measure µ̂N 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 Corollary3.2). The existence and uniqueness problems for this limit law are not obvious and are the analogue here of the existence and uniqueness problem forasymmetric 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 δQ , which entails quenched laws of largenumbers.We finally underline how our method can be used to study the evolution ofthe Gibbs measure µNJ 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 stochasticdifferential equation.The organization of the paper is as follows.In Section 2, we state and prove the strong large deviation upper bound.For more detail, see the following.1. In Section 2.1, we introduce the rate function and state the strong largedeviation upper bound (see Proposition 2.2 and Theorem 2.3).2. In Section 2.2, we prove that the law of the path space dynamics averagedon the couplings is absolutely continuous with respect to the law of thesedynamics with no couplings and show that its Radon–Nikodym derivativeis a function of the empirical measure.3. In Section 2.3, we study the continuity properties of this density.4. In Section 2.4, these continuity properties enable us to prove that the ratefunction is a good rate function in the short time or high temperatureregime.5. In Section 2.5, we prove the strong large deviation upper bound in the shorttime or high temperature regime by first proving an exponential tightnessresult and then a weak large deviation upper bound.In Section 3, we study the minima of the good rate function and provethat 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 Theorem3.14. This gives a propagation of chaos result stated in Corollary 3.3. In orderto give a hint about what kind of result this approach leads to, let us state

1370G. BEN AROUS AND A. GUIONNEThere Corollary 3.3.(ii): For any bounded continuous functions f1 fm onC 0 T A A m lim fi x dQ x f1 x1 · · · fm xm dPβN J x N i 1where is the expectation over the Gaussian couplings.In Section 3.1, we characterize the minima of the good rate function.In Section 3.2, we reduce the problem of finding these minima to a fixedpoint problem and then we show that this fixed point problem has at most onesolution.In Section 4, we apply our strategy to the stationary law of spin glassdynamics starting from the Gibbs measure. To this end, we need to supposethat β is small enough so that we are below the phase transition and that thefree energy concentrates as proved by Talagrand (see [17]). Then, the study ofthe law of the empirical measure is reduced to that of the law of the empiricalNmeasure starting from the nonnormalized Gibbs measure ZNJ µJ , which canbe studied following the above procedure. We then describe the asymptoticbehavior of the empirical measure.2. Averaged and quenched large deviation upper bounds.2.1. Statement of the large deviation upper bound. We first make precisethe setting of our model: let A be a strictly positive real and U be a C2 function on the interval A A such that U tends to infinity, when x A,sufficiently fast to insure that x ylimexp 2U y exp 2U z dz dy x A 00For any number N of particles, any temperature ( 1/β) and J Jij 1 i j N RN N , we consider the following system βN J of interacting diffusions. For j 1 N , N β dxjt U xjt dt dBjt Jji xit dt N i 1 βN J Law of x0 µ N0 where Bj 1 j N is an N-dimensional Brownian motion and µ0 is a probabilitymeasure on A A which does not put mass on the boundary A A .Under these assumptions, we recall Proposition 2.1 of [3].Proposition 2.1. For each J RN N , βN J has a unique weak solutionjand, almost surely, sups T sup1 j N xs does not reach A.In the following pages, we will focus on the evolution of this dynamicalsystem until a time T and denote by PβN J the weak solution of βN J

1371SYMMETRIC LANGEVIN SPIN GLASS DYNAMICSrestricted to the sigma algebra T σ xis 1 i N s T , and by P Nthe weak solution of 0N J restricted to T .Let WAT be the space of continuous functions from 0 T into A A . Then N Proposition 2.1 insures that PβN J is a probability measure on WA.TWe now suppose that the Jij ’s are random and that their distribution isgiven by the following.1. For any integer numbers i j , Jij Jji .2. If i j, the Jij ’s are independent centered Gaussian variables with covariance 1.3. The Jii ’s are independent centered Gaussian variables with covariance 2.They are also independent of the Jij i j .We shall denote by γ the law of the Jij ’s and by expectation under γ. Wehave already noticed in [3] that PβN J is a measurable function of the Jij ’s.Further, we will be interested in the averaged law QNβ:QNβ PβN J ω dγ ω The aim of this section is to prove that the law of the empirical measure underQNβ satisfies a large deviation upper bound, which entails a quenched largedeviation upper bound. To this end, we first define the rate function H whichgoverns this upper bound (see Proposition 2.2). In order to define H, we needsome notation and definitions that will also be useful later.1. Let µ 1 WAT T0 U xs ds2 dµ x 2. Let µ be a probability measure in . We denote by L2µ WAT the space ofA2 W isaHilbert spacethe square integrable functionsunderµ.HenceLµT with scalar product f g µ gf dµ.3. Let I be the identity on L2µ WAT .4. Let T be an integral operator on L2µ WAT with kernelbT x y T0xt yt dt Then T is a symmetric nonnegative Hilbert–Schmidt operator in L2µ WAT [for any µ 1 WA ].T5. Let λi be the eigenvalues of T in L2µ WAT , and Ei i N be an orthonormalbasis of eigenvectors of T such that T Ei λi Ei . Since T is nonnegative, the λi ’s are nonnegative so that we can define a symmetric positiveHilbert–Schmidt operator log I β2 T in L2µ WAT by i N log I β2 T Ei log 1 β2 λi Ei

1372G. BEN AROUS AND A. GUIONNET6. We define another integral operator T with kernel T xs U ys ds aT x y 12 xT yT x0 y0 0T0ys U xs ds ThenHilbert–Schmidt operator in L2µ WAT , since T T is a symmetric2 0 U xs ds dµ is finite.7. We denoteby trµ the trace in L2µ WAT . 8. Let I P be the relative entropy with respect to P: dµ logdµ if µ P dPI µP otherwise Proposition 2.2 (Definition). We can define a map * from into R by 2 λ1(2) * µ trµ log I β2 T dλtrµ T exp λ T exp2β20and a map H from 1 WAT into R by I µ P * µ H µ if I µ P otherwise.Proof. We first show that * is well defined and finite for any µ in [see(11) too].Indeed, as T is a nonnegative Hilbert–Schmidt operator, trµ log I β2 T is well defined and is finite according to (11) for any µ 1 WAT .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 traceclass. Further, since T is nonnegative, trµ T exp λ T 2 trµ T 2 . So,for any µ in , the second term in the right-hand side of (2) exists and isbounded.we will see later (see Lemma A.8) that, when I µ P is finite, Moreover,T 0 U xs ds 2 dµ is finite so that µ 1 WAT /I µ P .Thus, H is well defined and finite on µ 1 WA /I µ P . TWe shall prove the following theorem.Theorem 2.3. If 2β2 A2 T 1, then we have the following:(i) H is a good rate function; that is, H takes its values in 0 and, forall M R, H M is a compact subset of 1 WAT .(ii) For any closed subset F of 1 WAT ,lim supN N11 δ i F inf H log QNβFNN i 1 x

SYMMETRIC LANGEVIN SPIN GLASS DYNAMICS1373From Theorem 2.3, we can deduce the following quenched large deviationupper bound as in [3].Theorem 2.4. If 2β2 A2 T 1, for any closed subset F of 1 WAT and foralmost all J,lim supN N11 log PβN J δ i F inf H FNN i 1 xWe omit the proof that Theorem 2.3 implies Theorem 2.4 since it parallelsthe proof given in [3], Appendix C. The strategy of the proof of Theorem 2.3is the following.1. First, we prove (see Section 2.2) that QNβ is absolutely continuous withrespect to P N and that the Radon–Nikodym derivative of QNβ with respect Nto Pis equal, in the large deviation scaling, to exp N* µ̂N . Hence,according to Laplace-type methods, Theorem 2.3(ii) is not surprising (see[2] and [7]).2. Once we are motivated by this last result, we study H and prove that it isa 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.N2.2. Study of QNβ . We first show that Qβ is absolutely continuous withrespect to P N and give the Radon–Nykodim derivative of QNβ with respectto P N .The Girsanov theorem implies that, for almost all couplings J, PβN J isabsolutely continuous with respect to P N and describes its Radon–Nikodymderivative. Thus, it is not hard to see that, if we denote by Bi the processtdefined by Bit Bt xi xit xi0 0 U xis ds, then NN TNdPβ J 1 jNMβ T expβJji xit dBt dP N0N i 1j 1(3) 2Nβ2 T1 i Jji xt dt 2 0N i 1and we have the following proposition. NandProposition 2.5. We have QNβ PdQNβdP N MNβ T In order to study the law of the empirical measure under QNβ , we want toprove that MNβ T is a function of the empirical measure. More precisely, let IA2be the identity in the tensor product space L2µ WAT Lµ WT and trµ µ the

1374G. BEN AROUS AND A. GUIONNETA2trace in L2µ WAT Lµ WT . We then define1 I β2 T I β2 I T * µ trµ µ log4 I β2 T I β2 T 1β2 Ttrµ log I 2β2 T β2 T trµ I 2β2 T 1 T 441 NDenote in short µ̂N for the empirical measure Ni 1 δxi .We are going to prove the following statement. Theorem 2.6.(i) We have, P N almost surely,NNMNβ T exp N* µ̂ * µ̂ (ii) There exists a finite constant C C β T A such that, for any discrete Aprobability measure on WAT , µ 1 WT , if dim µ denotes the dimension of2the image of T in L µ , * µ 1 C 1 dim µ 1/2 * µ 1 2so, if D exp C, P N almost surely, dQNCCβ1 NN DexpN1 D 1 N exp N 1 * µ̂N * µ̂ dP NNN TRemark. It is obvious that 0 U xs ds 2 dP x is finite. Hence, T T i2 0 U xs ds 2 dµ̂N x 1/N Nis P N almosti 1 0 U xs ds N NNalmost surely. Thus, * µ̂ is well defined,surely finite, that is, µ̂ , PP N almost surely.To prove Theorem 2.6, we shall use spectral theory.2.2.1. Spectral calculus. In the following pages, an integer N will be given.We may regard J Jij 1 i j N as an element of the space N of the N TN real symmetric matrices. For any x1 xN such that 0 U xis ds isfinite for any i 1 N , we define two other symmetric matrices A andB in RN N by T T1jjAij xit dBt xt dBit02 N 0 T T1jj xis U xjs ds xjs U xis ds δij T xiT xT xi0 x0 002 N1 T i jBij x x dt N 0 t tLet λi be the eigenvalues of B and ei be the eigenvectors of B in RN such thatBei λi ei . We prove the following.

1375SYMMETRIC LANGEVIN SPIN GLASS DYNAMICSProposition 2.7. We have, P N almost surely, NN e i Aej 21 NMβ T exp β2 log 1 β2 λi β2 λj 2 λ β2 λ1 β4iji j 1i j 1 N1 2 log 1 2β λi 4 i 1Proof. If tr denotes the trace in N , since J J , we get TNN 1 jJjixit dBt Aij Jji tr AJ 0N i j 1i j 1 TN1 Jji Jjkxit xkt dt tr JBJ tr JBJ N i j k 10So, since denotes the expectation with respect to the Gaussian variable J, Twe get that, for any x x1 xN such that 0 U xis ds is finite for1 i N and so P N almost surely, 1 2MNβ T exp β tr JA 2 β tr JBJ Using the usual rules of computation for Gaussian variables (see [13], Proposition 8.4), we get 1 2NMβ T exp β tr JBJ 2(4) 21 22 exp 1/2 β tr JBJ exp β tr JA 2 exp 1/2 β2 tr JBJ Lemma 2.8. exp 12 β2 tr JBJ N exp 41log 1 β2 λi λj i j 114N i 12 log 1 2β λi Proof. We have chosen the Jij 1 i j N ’s so that their law is invariant byrotation on RN ; that is, for any orthogonal matrix O, the law of Jij 1 i j Nis invariant by the action J OJO . Thus, if O is an orthogonal matrix suchthat OBO is a diagonal matrix D diag λ1 λN , then exp 21 β2 tr JBJ exp 12 β2 tr JDJ NN exp 41log 1 β2 λi λj 41log 1 2β2 λi i j 1i 1

1376G. BEN AROUS AND A. GUIONNETLemma 2.9. N e i Aej 2exp 1/2 β2 tr JBJ tr JA 2 2 exp 1/2 β2 tr JBJ 1 β2 λi λj i j 1 OAO . Since the law of J is invariant by rotation,Proof. Let A exp 1/2 β2 tr JBJ tr JA exp 1/2 β2 tr JBJ 22 exp 1/2 β tr JDJ tr JA exp 1/2 β2 tr JDJ exp 1/2 β2 tr JDJ A J JA ij klji lk exp 1/2 β2 tr JDJ ijkl 2However, exp 1/2 β2 tr JDJ Jij Jkl exp 1/2 β2 tr JDJ 0 if j i k l and l k 1 if j i k l or l k i j 2 1 β λi λj 2 if i j k l 1 β2 λi λj A , we concludeSince A 2N2A ij2 exp 1/2 β tr JBJ 2 tr JA 2 λ λ exp 1/2 β2 tr JBJ 1 βiji j 1Finally, according to the definition of O, if ei is the eigenvector of B associated e Ae , so we have proved Lemma 2.9. with the eigenvalue λi , then AijjiAccording to (4), Lemmas 2.8 and 2.9 give Proposition 2.7.2.2.2. Proof of Theorem 2 6. We shall now use Proposition 2.7 to expressMNβ T as a function of the empirical measure (and of N). To this end, we shallNare isomorphic whenever the xi ’s are distinct, anduse that L2µ̂N WAT

Ecole Normale Superieure and Universit´e 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 tem-