Year: 2025
Type: preprint
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.
Source: arXiv (Cornell University)
Authors: Faggionato, Alessandra, Hartarsky, Ivailo
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4415345579
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