Harnack Inequalities and Local Central Limit Theorem for the Polynomial Lower Tail Random Conductance Model

TL;DR

This paper establishes sharp bounds and fundamental inequalities for random walks with i.i.d. conductances having polynomial lower tails, leading to local CLTs and heat kernel estimates, applicable on general graphs.

Contribution

It provides new sharp bounds and inequalities for random walks with polynomial lower tail conductances, extending to general graphs and both constant and variable speed models.

Findings

Sharp transition probability bounds for conductance models

Derivation of local central limit theorems and Gaussian heat kernel bounds

Applicability of methods to general graph structures

Abstract

We prove upper bounds on the transition probabilities of random walks with i.i.d. random conductances with a polynomial lower tail near $0$. We consider both constant and variable speed models. Our estimates are sharp. As a consequence, we derive local central limit theorems, parabolic Harnack inequalities and Gaussian bounds for the heat kernel. Some of the arguments are robust and applicable for random walks on general graphs. Such results are stated under a general setting.