A central limit theorem with applications to percolation, epidemics and boolean models

Abstract

Suppose X = (Xx)x∈ℤd is a white noise process, and H(B), defined for finite subsets B of ℤd, is determined in a stationary way by the restriction of X to B. Using a martingale approach, we prove a central limit theorem (CLT) for H as B becomes large, subject to H satisfying a "stabilization" condition (the effect of changing Xx at a single site needs to be local). This CLT is then applied to component counts for percolation and Boolean models, to the size of the big cluster for percolation on a box, and to the final size of a spatial epidemic.