Crossings and diffusion in Poisson driven marked random connection models

TL;DR

This paper analyzes crossing statistics and diffusion properties in Poisson-driven random connection models, establishing exponential tail bounds and demonstrating non-degeneracy of effective diffusion matrices in various stochastic processes.

Contribution

It introduces new exponential tail bounds for crossings in RCM and proves non-degeneracy of effective diffusion matrices in related stochastic models.

Findings

Exponential tail bounds for crossings in supercritical RCM.

Non-degeneracy of effective homogenized matrices in random walks and resistor networks.

Application to Mott variable range hopping models.

Abstract

We first study crossing statistics in random connection models (RCM) 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, and percolation in slabs, in analogy with the Grimmett-Marstrand theorem. We then present several applications to transport and diffusion phenomena. In particular, we prove the non-degeneracy of the effective homogenized matrix arising in the large-scale limit of random walks, exclusion processes, and resistor networks on the RCM, and the non-degeneracy of the effective diffusion constant for one-dimensional diffusion operators on the Euclidean graph associated with the RCM. As examples, we apply our results to Poisson-Boolean models and Mott variable range hopping random resistor network, providing a fundamental ingredient used in the derivation of Mott's law.