Crossings and diffusion in Poisson driven marked random connection models

2025/07/05 by Alessandra Faggionato, Faggionato, Alessandra, Ivailo Hartarsky +1 · 2 citations
Computer Science · Physics and Astronomy · #60D05 #60K35 #60K37 #82B21 #Complex Network Analysis Techniques #Data Management and Algorithms #FOS: Mathematics #Probability (math.PR) #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.2507.03965

openalex publication_date 2025/07/05 · openalex created_date 2025/10/20 · openalex updated_date 2026/07/28

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.

Citations

• Supercritical phase of the random connection model
• On the uniqueness of the infinite cluster and the cluster density in the Poisson driven random connection model
• Quantitative homogenization and large-scale regularity of Poisson point clouds
• Mott's law for the v.r.h. random resistor network and for Mott's random walk
• Almost sharp sharpness for Poisson Boolean percolation
• Supercritical percolation on graphs of polynomial growth
• Hydrodynamic limit of simple exclusion processes in symmetric random environments via duality and homogenization
• Stochastic homogenization of random walks on point processes
• Recurrence versus Transience for Weight-Dependent Random Connection Models
• Upper bounds on the percolation correlation length
• Existence and continuity of the flow constant in first passage percolation
• Elliptic regularity and quantitative homogenization on percolation clusters
• On the Ornstein-Zernike equation for stationary cluster processes and the random connection model
• Is the critical percolation probability local?
• Limit theorems for maximum flows on a lattice
• Random walks on supercritical percolation clusters
• Random Walks and Electric Networks

Cited by

• Crossing probabilities in geometric inhomogeneous random graphs
• Supercritical phase of the random connection model

Related

• Connection probabilities in Poisson random graphs with uniformly bounded\n edges
• Left-right crossings in the Miller-Abrahams random resistor network and in generalized Boolean models
• Critical Exponents for Marked Random Connection Models
• Random walks and exclusion processes among random conductances on random infinite clusters: homogenization and hydrodynamic limit
• Stochastic homogenization in amorphous media and applications to exclusion processes