A Central Limit Theorem With Applications to Percolation, Epidemics and Boolean Models

Abstract

Suppose $X = (X_x)_{x\in\mathbb{Z}^d}$ is a white noise process, and $H (B)$, defined for finite subsets $B$ of $\math{Z}^d$, 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 $X _x$ 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.