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Mean Field Models

Hopfield Models as Generalized Random Mean Field Models

Twenty years ago, Pastur and Figotin [FP1,FP2] first introduced and studied what has become known as the Hopfield model and which turned out, over the years, to be one of the more successful and important models of a disordered system. This is also reflected in the fact that several contributions in this book are devoted to it. The Hopfield model is quite versatile and models various situations. Pastur and Figotin introduced it as a simple model for a spin glass, and, in 1982, Hopfield independently considered it as a model for associative memory.
Anton Bovier, Véronique Gayrard

The Martingale Method for Mean-Field Disordered Systems at High Temperature

We review some martingale techniques for the study of meanfield disordered systems at high temperature, including the Sherrington-Kirkpatrick model, the Hopfield model, and more general associative memory models. We describe the log-normal limit of the partition function in term of a Brownian motion, and we give another proof of self-averaging for pressure. In the Sherrington-Kirkpatrick case we show that the law of the product of two independent configurations is close to uniform at very high temperature, but this does not persist in the whole high-temperature region.
Francis Comets

On the Central Limit Theorem for the Overlap in the Hopfield Model

We consider the Hopfield model with N neurons and an increasing number M = M(N) of randomly chosen patterns. Under the condition M 2/N → 0, we prove for every fixed choice of overlap parameters a central limit theorem as N → ∞, which holds for almost all realizations of the random patterns. In the special case where the temperature is above the critical one and there is no external magnetic field, the condition M 3/2 log MN suffices. As in the case of a finite number of patterns, the central limit theorem requires a centering which depends on the random patterns. In addition, we describe the almost sure asymptotic behavior of the partition function under the condition M 3/N → 0.
Barbara Gentz

Limiting Behavior of Random Gibbs Measures: Metastates in Some Disordered Mean Field Models

We present examples of random mean field spin models for which the size dependence of their Gibbs measures μΛ n can be rigorously analyzed. We investigate their ‘empirical metastates’ \(1/N \sum\nolimits_{n = 1}^N {\delta _{\mu \Lambda _n } }\), introduced by Newman and Stein, along the sequence of finite volumes \(\Lambda _n = \{ 1,\,...,\,n\}\). The empirical metastate is shown not to converge in our examples if the realization of the disorder is fixed. This phenomenon leads us to consider the distributions w.r.t disorder of the empirical metastates for which we show convergence and give explicit limiting expression.
C. Külske

On the Storage Capacity of the Hopfield Model

We give a review on the rigorous results concerning the storage capacity of the Hopfield model. We distinguish between two different concepts of storage both of them guided by the idea that the retrieval dynamics is a Monte Carlo dynamics (possibly at zero temperature). We recall the results of McEliece et al. [MPRV87] as well as those by Newman [N88] for the storage capacity of the Hopfield model with unbiased i.i.d. patterns and comprehend some recent development concerning the Hopfield model with semantically correlated or biased patterns.
Matthias Löwe

Lattice Models

Typical Profiles of the Kac-Hopfield Model

Mean field models, random or not, are very important for explaining simply the general phenomenon of phase transitions. However, for random systems, in general, their analysis, as many of the contributions in this volume confirm, is not simple at all, a fact which may justify the amount of effort spent on them. In spite of all that, mean fields models, in many respects, are only poor caricatures of realistic systems1 and are unable to feature some of their most important aspects; in particular, in a phase-transition regime, they are unable to properly account for the phenomenon of phase separation, that is, the observed feature that states of the system with two or more phases coexist in separate regions of space. This deficiency manifests itself also in the fact that the canonical free energy is generally not a convex function of the order parameters, which in turn means that the usual formalism of thermodynamics cannot be immediately used (e.g., the isotherms are not monotone, thus cannot directly be used to determine the equations of state, and insisting on doing so would produce a totally unphysical effect, like regions of parameters where pressure is a decreasing function of density). This problem is solved by the Maxwell construction, by which free energy is simply replaced ad hoc by its convex hull.
A. Bovier, V. Gayrard, P. Picco

Thermodynamic Chaos and the Structure of Short-Range Spin Glasses

This paper presents an approach, recently introduced by the authors and based on the notion of “metastates,” to the chaotic size dependence expected in systems with many competing pure states and applies it to the Edwards-Anderson (EA) spin glass model. We begin by reviewing the standard picture of the EA model based on the Sherrington-Kirkpatrick (SK) model and why that standard SK picture is untenable. Then we introduce metastates, which are the analogues of the invariant probability measures describing chaotic dynamical systems and discuss how they should appear in several models simpler than the EA spin glass. Finally, we consider possibilities for the nature of the EA metastate, including one which is a nonstandard SK picture, and speculate on their prospects. An appendix contains proofs used in our construction of metastates and in the earlier construction by Aizenman and Wehr.
Charles M. Newman, Daniel L. Stein

Random Spin Systems with Long-Range Interactions

We present the state of the matter for random spin systems with long-range interactions.
Bogusław Zegarlinski



Langevin Dynamics for Sherrington-Kirkpatrick Spin Glasses

The purpose of this note is to review and summarize recent results on dynamics of mean-field spin glasses.
Gerard Ben Arous, Alice Guionnet

Sherrington-Kirkpatrick Spin-Glass Dynamics

Part II: The Discrete Setting
In this second part we want to describe the Glauber dynamics approach for the Sherrington-Kirkpatrick spin-glass dynamics. The physical motivation we have in mind is the same as described by Ben Arous and Guionnet in the proceedings for the “continuous setting.” Glauber dynamics means that we want to study a reversible Markov process directly for the Sherrington-Kirkpatrick (SK) Gibbs measures on the (discrete) state space {−1, 1} N , that is, in the “hard” spin picture. To formulate such dynamics, we have to use jump processes. The use of jump processes is really more natural in the SK model than the use of diffusions, although diffusions are closer to physical intuition. This jump process ansatz was introduced in [Som87]. Some rigorous results for asymmetric dynamics were proved in [Gru92]. There is a strong advantage in the use of Glauber dynamics. The Girsanov exponent, used to describe the interacting model, is fairly well behaved, which permits proving large deviation results without restriction on time and temperature. These large deviation results are strong enough to guaranty direct convergence results to limiting dynamics. As an additional present from exponential bounds, cpnvergence holds for almost all realizations of couplings J. The price we have to pay for the use of Glauber dynamics is the loss of Gaussian techniques, which makes it necessary to consider an enlarged state space (we include the “fields” in our picture), which in the end gives a more transparent description of limiting dynamics.
M. Grunwald


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