Quenched invariance principle for a long-range random walk with unbounded conductances

TL;DR

This paper proves a quenched invariance principle for a long-range, reversible random walk on a percolation cluster with unbounded conductances, using the corrector method and metric comparisons.

Contribution

It establishes the quenched invariance principle for long-range random walks with unbounded conductances on percolation clusters, extending previous results to more general environments.

Findings

Proved the quenched invariance principle for the model.

Established metric comparison between graph and Euclidean metrics.

Derived quenched heat kernel estimates.

Abstract

We consider a random walk on a random graph $(V,E)$, where $V$ is the set of open sites under i.i.d. Bernoulli site percolation on the multi-dimensional integer set $\mathbf{Z}^d$, and the transition probabilities of the walk are generated by i.i.d. random conductances (positive numbers) assigned to the edges in $E$. This random walk in random environments has long range jumps and is reversible. We prove the quenched invariance principle for this walk when the random conductances are unbounded from above but uniformly bounded from zero by taking the corrector approach. To this end, we prove a metric comparison between the graph metric and the Euclidean metric on the graph $(V, E)$, an estimation of a first-passage percolation and an almost surely weighted Poincar{\'{e}} inequality on $(V,E)$, which are used to prove the quenched heat kernel estimations for the random walk.