Stochastic models for large interacting systems and related correlation inequalities.

Abstract

A very large and active part of probability theory is concerned with the formulation and analysis of models for the evolution of large systems arising in the sciences, including physics and biology. These models have in their description randomness in the evolution rules, and interactions among various parts of the system. This article describes some of the main models in this area, as well as some of the major results about their behavior that have been obtained during the past 40 years. An important technique in this area, as well as in related parts of physics, is the use of correlation inequalities. These express positive or negative dependence between random quantities related to the model. In some types of models, the underlying dependence is positive, whereas in others it is negative. We give particular attention to these issues, and to applications of these inequalities. Among the applications are central limit theorems that give convergence to a Gaussian distribution.