Probability Theory and Spin Glasses.

Ohio State University Research Foundation -Do Not Use, Columbus OH

Investigators

Abstract

0243813 Talagrand Mean field models for spin glasses (such as the famous Sherrington Kirkpatrick model) exhibit some completely new phenomenon of probability theory, that at a high level can be described as the existence of intricate structures among the near extreme values of large families of correlated random variables. The rigorous study of these models is still in its infancy, and the main goal of this proposal is to pursue it. Two of the specific goals are to obtain a better understanding of the low-temperature phase of the p-spin interaction model, and to start the study of the new Hamiltonians introduced by F. Guerra in his broken replica-symmetry bound for the Sherrington Kirkpatrick model. One of the central themes of probability theory is the emergence of a kind of order out of large number of random events. This is expressed by classical theorems, such as the law of large numbers. These classical results relate to situations where the random events are independent, that is, the outcome of one does not influence the outcome of the others. Such an assumption is unrealistic in practice. We investigate the emergence of new types of collective behavior in large collections of random events, where each event can, and will, influence the outcome of the others. The specific mathematical models have been motivated by the behavior of magnetized particles, and are known as "spin glasses".