Quenched invariance principles for the random conductance model on a random graph with degenerate ergodic weights

TL;DR

This paper proves a quenched invariance principle for random walks on a random graph with ergodic, possibly degenerate conductances, using a new isoperimetric inequality, advancing understanding of diffusive behavior in disordered media.

Contribution

It establishes a quenched invariance principle for the random conductance model on a random graph with degenerate ergodic weights, under mild conditions, introducing a novel anchored isoperimetric inequality.

Findings

Proved quenched invariance principle for the model.

Established a new anchored relative isoperimetric inequality.

Demonstrated diffusive behavior in the presence of degenerate conductances.

Abstract

We consider a stationary and ergodic random field $\{\omega(e) : e \in E_d\}$ that is parameterized by the edge set of the Euclidean lattice $\mathbb{Z}^d$, $d \geq 2$. The random variable $\omega(e)$, taking values in $[0, \infty)$ and satisfying certain moment bounds, is thought of as the conductance of the edge $e$. Assuming that the set of edges with positive conductances give rise to a unique infinite cluster $\mathcal{C}_{\infty}(\omega)$, we prove a quenched invariance principle for the continuous-time random walk among random conductances under relatively mild conditions on the structure of the infinite cluster. An essential ingredient of our proof is a new anchored relative isoperimetric inequality.