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

Zhang Zhongyang, Zhang Li-Xin

Published 2013-11-21, updated 2016-03-11Version 3

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.