The critical two-point function for long-range percolation on the hierarchical lattice

Abstract

We prove up-to-constants bounds on the two-point function (i.e., point-to-point connection probabilities) for critical long-range percolation on the d-dimensional hierarchical lattice. More precisely, we prove that if we connect each pair of points x and y by an edge with probability 1-exp(-β||x-y||^(-d-α)), where 0 < α < d is fixed and β ≥ 0 is a parameter, then the critical two-point function satisfies P_(β_c)(x ↔ y)||x-y||^(-d+α) for every pair of distinct points x and y. We deduce in particular that the model has mean-field critical behaviour when α < d/3 and does not have mean-field critical behaviour when α > d/3.