The RSW Theorem for Continuum Percolation and the CLT for Euclidean Minimal Spanning Trees

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 . in [ 0, 1]2 as . . .. 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.