Multivariate spatial central limit theorems with applications to percolation and spatial graphs

Abstract

Suppose $X = (X_x, x$ in $Z^d)$ is a family of i.i.d. variables in some measurable space, $B_0$ is a bounded set in $R^d$, and for $t > 1$, $H_t$ is a measure on $tB_0$ determined by the restriction of $X$ to lattice sites in or adjacent to $tB_0$. We prove convergence to a white noise process for the random measure on $B_0$ given by $t^{-d/2}(H_t(tA)-EH_t(tA))$ for subsets $A$ of $B_0$, as $t$ becomes large,subject to $H$ satisfying a ``stabilization'' condition (whereby the effect of changing $X$ at a single site $x$ is local) but with no assumptions on the rate of decay of correlations. We also give a multivariate central limit theorem for the joint distributions of two or more such measures $H_t$, and adapt the result to measures based on Poisson and binomial point processes. Applications given include a white noise limit for the measure which counts clusters of critical percolation, a functional central limit theorem for the empirical process of the edge lengths of the minimal spanning tree on random points, and central limit theorems for the on-line nearest neighbour graph.