Heat-kernel estimates for random walk among random conductances with heavy tail

TL;DR

This paper investigates heat-kernel decay in random walks on $ ext{Z}^d$ with conductances having heavy tails near zero, revealing anomalous decay behavior in high dimensions and near-standard decay for large tail exponents.

Contribution

It establishes new decay estimates for the heat kernel in models with heavy-tailed conductances, highlighting the transition from anomalous to standard decay as tail heaviness varies.

Findings

Return probability decay is non-Gaussian and approaches a constant times $n^{-2}$ as tail exponent $eta$ approaches zero.

Heat-kernel decay can be made arbitrarily close to the standard $n^{-d/2}$ decay in a logarithmic sense for large $eta$.

Results hold for all dimensions $d geq 5$.

Abstract

We study models of discrete-time, symmetric, $\Z^{d}$-valued random walks in random environments, driven by a field of i.i.d. random nearest-neighbor conductances $\omega_{xy}\in[0,1]$, with polynomial tail near 0 with exponent $\gamma>0$. We first prove for all $d\geq5$ that the return probability shows an anomalous decay (non-Gaussian) that approches (up to sub-polynomial terms) a random constant times $n^{-2}$ when we push the power $\gamma$ to zero. In contrast, we prove that the heat-kernel decay is as close as we want, in a logarithmic sense, to the standard decay $n^{-d/2}$ for large values of the parameter $\gamma$.