The RSW theorem for continuum percolation and the CLT for Euclidean minimal spanning trees

Abstract

We prove a central limit theorem for the length of the minimal spanning tree of the set of sites of a Poisson process of intensity $\lambda$ in $[0, 1]^2$ as $\lambda \to \infty$. As observed previously by Ramey, the main difficulty is the dependency between the contributions to this length from different regions of $[0, 1]^2$; a percolation-theoretic result on circuits surrounding a fixed site can be used to control this dependency. We prove such a result via a continuum percolation version of the Russo-Seymour-Welsh theorem for occupied crossings of a rectangle. This RSW theorem also yields a variety of results for two-dimensional fixed-radius continuum percolation already well known for lattice models, including a finite-box criterion for percolation and absence of percolation at the critical point.