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, B₀ is a bounded set in R^d, and for t>1, H_t is a measure on tB₀ determined by the restriction of X to lattice sites in or adjacent to tB₀. We prove convergence to a white noise process for the random measure on B₀ given by t^−d/2(H_t(tA)−EH_t(tA)) for subsets A of B₀, 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-neighbor graph.