Critical Exponents for Marked Random Connection Models

Abstract

Here we prove critical exponents for Random Connections Models (RCMs) with random marks. The vertices are given by a marked Poisson point process on $\mathbb{R}^d$ and an edge exists between any pair of vertices independently with a probability depending upon their spatial displacement and on their respective marks. Given conditions on the edge probabilities, we prove mean-field lower bounds for the susceptibility and percolation functions. In particular, we prove the equality of the susceptibility and percolation critical intensities. If we assume that a form of the triangle condition holds, then we also prove that the susceptibility, percolation and cluster tail critical exponents exist and take their mean-field values. Our proof approach adapts the differential inequality and magnetization function approaches that have been previously applied to discrete homogeneous settings to our continuum marked setting. This includes a proof of the analyticity of the magnetization function in the required parameter regime.