Spin Glasses: A New Direction for Probability Theory

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

Investigators

Abstract

9988480 Talagrand While studying mean field models for spin glasses (such as the famous Sherrington Kirkpatrick model) a number of physicists "discovered" (or more accurately, conjectured) some completely new phenomenon of probability theory, namely the existence of intricate structures among the near extreme values of large families of correlated random variables. The main goal of this research is, through the study of the main models, to provide mathematically rigorous foundations for their conjectures. Two main objectives are the control of the entire high temperature region for the main models, and a complete description of the low temperature phase of the simplest models. 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 the main classical theorems, starting with the law of large numbers. These classical result relate to situations where the events are independent, that is do not influence each other. 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 all the others.