Percolation and Minimal Spanning Trees

Abstract

Consider a random set $V_n$ of points in the box [n, −n)^d, generated either by a Poisson process with density p or by a site percolation process with parameter p. We analyze the empirical distribution function F _n of the lengths of edges in a minimal (Euclidean) spanning tree $T_n$ on $V_n$. We express the limit of F _n, as n → ∞, in terms of the free energies of a family of percolation processes derived from $V_n$ by declaring two points to be adjacent whenever they are closer than a prescribed distance. By exploring the singularities of such free energies, we show that the large-n limits of the moments of F _n are infinitely differentiable functions of p except possibly at values belonging to a certain infinite sequence (p _c(k): k ≥ 1) of critical percolation probabilities. It is believed that, in two dimensions, these limiting moments are twice differentiable at these singular values, but not thrice differentiable. This analysis provides a rigorous framework for the numerical experimentation of Dussert, Rasigni, Rasigni, Palmari, and Llebaria, who have proposed novel Monte Carlo methods for estimating the numerical values of critical percolation probabilities.