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.contact process | exclusion process | Glauber dynamics | voter models Models for Interacting SystemsDuring the past half century, mathematical models for the evolution of large interacting systems arising in a number of scientific areas have been proposed and analyzed. Here are some of these areas, together with a sampling of the many papers and books in which such models have been discussed: magnetic systems (1), high energy scattering (2), dynamics of mutation in a structured population (3), tumor growth (4, 5), competition between different strains of viruses (6), mutations of pathogens (7), biopolymers (8), epidemics (9, 10), ecology (11, 12), hydrology (13), cooperative behavior (14-16), spatial distribution of unemployment (17), and the analysis of traffic flow (18)(19)(20).The main objective in the study of these models is to describe their long-time behavior. Usually, the models contain one or more parameters. An important issue is to determine how the long-time behavior depends on these parameters. Often there is a sharp transition in the nature of the behavior at some particular parameter value. This situation is described by saying that a phase transition occurs there.Some of the analysis of these systems has been mathematical, whereas other approaches have been based on simulations and heuristics. Dobrushin (21) and Spitzer (22) are usually credited with initiating the mathematical developments about 40 years ago. The modern theory of models of this type is treated in my two monographs (23,24). Typically, the model is a random process η t with state space f0;1g Z d of binary configurations on the d-dimensional integer lattice Z d . The interpretation of the values 0 and 1 at a site x ∈ Z d depends on the model, and on the area that motivated it. The process satisfies the Markov property, which means that once one knows the state of the system at a given time t, the evolution of the system after that time does not depend on its behavior before time t. It follows that the evolution rules can be described by specifying how the proc...

Kawasaki Dynamics in Continuum: Micro- and Mesoscopic Descriptions

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