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Crossings and diffusion in Poisson driven marked random connection models

Alessandra Faggionato, Ivailo Hartarsky

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.

Crossings and diffusion in Poisson driven marked random connection models

Abstract

We first study crossing statistics in random connection models (RCM) built on marked Poisson point processes on . 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.

Paper Structure

This paper contains 23 sections, 23 theorems, 78 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Crossings and transport
  3. Main results
  4. Proof outline and comments on the strategy
  5. Model and main results
  6. Random connection model crossings
  7. Transport via resistor networks
  8. Transport via diffusions on singular geometric structures
  9. Application to generalized Poisson--Boolean models
  10. Poisson--Boolean model
  11. Mott variable range hopping
  12. Preliminaries
  13. RCM driven by a PPP on RdxM
  14. FKG inequality
  15. Irreducibility
  16. ...and 8 more sections

Key Result

Theorem 2.4

Suppose the triple $(\lambda,\nu,\varphi)$ satisfies Assumptions assumere. Then for any $\rho>\lambda$ there exist $c_1,c_2>0$ such that, for all $\ell$ large enough, we have where $\mathcal{N}_{\ell}$ denotes the maximal number of vertex-disjoint LR crossings of $\Lambda_{\ell}$ included in the unique infinite cluster of $G_d$.

Figures (1)

  • Figure 1: Illustration of Definition \ref{['def:occupied']}. The last $d-2$ dimensions and marks are suppressed.

Theorems & Definitions (58)

  • Definition 2.1: Graph $G_d$
  • Definition 2.2: LR crossing
  • Theorem 2.4: Lower bounds on LR crossings
  • Remark 2.5
  • Theorem 2.6: Percolation in slabs
  • Definition 2.7: Resistor network ${\rm RN}_{\ell}$
  • Definition 2.8: Directional conductivity $\sigma_{\ell}$
  • Lemma 2.9: Lower bound on $\sigma_{\ell}$
  • proof
  • Proposition 2.10: Faggionato25*Corollary 2.7
  • ...and 48 more