Anomalous Heat Kernel for Random Walks in Random Environments of Conductances

TL;DR

This paper investigates the anomalous heat kernel decay in random walks within random environments of conductances, revealing dimension-dependent behaviors and decay rates for conductances with polynomial tail distributions.

Contribution

It provides the first precise decay order of the heat kernel for conductances with polynomial tails near zero in dimensions five and higher.

Findings

Decay of heat kernel in $d \\ge 5$ with conductances in $[0,1]$

Normal behavior in $d=4$

Opposite return probability behavior with conductances in $[1,\\infty)$

Abstract

We study the trapping phenomenon of random walks in random environments of i.i.d. random conductances on the bonds of the grid $\mathbb{Z}^d$, the so-called random conductance model. Our main results concern the important model with conductances in $[0, 1]$ and a polynomial-tailed law near zero for which we find the correct order of decay of the anomalous heat kernel for $d \ge 5$. In $d = 4$, the behavior is found to be normal. In addition, we look at the symmetrical situation with conductances in $[1, \infty)$ with a polynomial law at infinity, which also shows opposite return probability behaviors.