Scaling limit of the directional conductivity of random resistor networks on simple point processes

TL;DR

This paper establishes the asymptotic behavior of directional conductivity in random resistor networks on point processes, showing convergence to eigenvalues of an effective matrix and covering various models including percolation and amorphous solids.

Contribution

It provides a general proof of the scaling limit of directional conductivity for a wide class of random resistor networks with unbounded filament lengths, extending previous results.

Findings

Directional conductivity converges to eigenvalues of the effective matrix D.

Results apply to models like lattice, continuum percolation, and amorphous solids.

Extends bounds consistent with Mott's law for conduction in disordered materials.

Abstract

We consider random resistor networks with nodes given by a point process on $\mathbb{R}^d$ and with random conductances. The length range of the electrical filaments can be unbounded. We assume that the randomness is stationary and ergodic w.r.t. the action of the group $\mathbb{G}$, given by $\mathbb{R}^d$ or $\mathbb{Z}^d$. This action is covariant w.r.t. translations on the Euclidean space. Under minimal assumptions we prove that a.s. the suitably rescaled directional conductivity of the resistor network along the principal directions of the effective homogenized matrix $D$ converges to the corresponding eigenvalue of $D$ times the intensity of the point process. More generally, we prove a quenched scaling limit of the directional conductivity along any vector $e\in {\rm Ker}(D) \cup {\rm Ker}(D)^\perp$. Our results cover plenty of models including e.g. the standard conductance model on $\mathbb{Z}^d$ (also with long filaments), the Miller-Abrahams resistor network for conduction in amorphous solids (to which we can now extend the bounds in agreement with Mott's law previously obtained in \cite{CP1,FM,FSS} for Mott's random walk), resistor networks on the supercritical cluster in lattice and continuum percolations, resistor networks on crystal lattices and on Delaunay triangulations.