On symmetric random walks with random conductances on $\Z^d$

TL;DR

This paper investigates symmetric random walks in random environments on ^d, deriving decay estimates for transition probabilities without uniform ellipticity, focusing on conductances with polynomial tail near zero.

Contribution

It provides new asymptotic results for return probabilities and convergence times for random walks with conductances having polynomial tails near zero, without assuming uniform ellipticity.

Findings

Derived decay estimates for transition probabilities.

Obtained asymptotics for annealed return probability.

Analyzed convergence times in finite boxes.

Abstract

We study models of continuous time, symmetric, $\Z^d$-valued random walks in random environments. One of our aims is to derive estimates on the decay of transition probabilities in a case where a uniform ellipticity assumption is absent. We consider the case of independent conductances with a polynomial tail near 0, and obtain precise asymptotics for the annealed return probability and convergence times for the random walk confined to a finite box.