Crossings and diffusion in Poisson driven marked random connection models (2507.03965v1)

Abstract: Motivated by applications to stochastic homogenization, we study crossing statistics in random connection models built on marked Poisson point processes on $\mathbb R^d$. Under general assumptions, we show exponential tail bounds for the number of crossings of a box contained in the infinite cluster for supercritical intensity of the point process. This entails the non-degeneracy of the homogenized diffusion matrix arising in random walks, exclusion processes, diffusions and other models on the RCM. As examples, we apply our result to Poisson-Boolean models and Mott variable range hopping random resistor network, providing a fundamental ingredient used in the derivation of Mott's law. The proof relies on adaptations of Tanemura's growth process, the Duminil-Copin, Kozma and Tassion seedless renormalisation scheme, Chebunin and Last's uniqueness, as well as new ideas.