PERCOLATION AND MINIMAL SPANNING-TREES

Consider a random set V-n of points in the box [n, -n)(d), generated e ither by a Poisson process with density p or by a site percolation pro cess 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 --> infinity, in terms of t he 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 ene rgies, we show that the large-n limits of the moments of F-n are infin itely differentiable functions of p except possibly at values belongin g to a certain infinite sequence (p(c)(k): k greater than or equal to 1) of critical percolation probabilities. It is believed that, in two dimensions, these limiting moments are twice differentiable at these s ingular values, but not thrice differentiable. This analysis provides a rigorous framework for the numerical experimentation of Dussert, Ras igni, Rasigni, Palmari, and Llebaria, who have proposed novel Monte Ca rlo methods for estimating the numerical values of critical percolatio n probabilities.