A Gentle Introduction to Cluster Expansions

Abstract

A measure on a high dimensional space is to be defined by a density with respect to a reference measure. The goal is to control this measure in the limit as the dimension becomes infinite. A cluster expansion is a means to accomplish this. The idea is to express expectations with respect to the reference measure as a combinatorial exponential of cluster coefficients. This is an elementary exposition of how the analysis of a discrete gas system in equilibrium can be used to obtain estimates on the expansion in cluster coefficients. The main topic is an analysis of the Mayer equations for the gas system that makes no use of graph combinatorics.