Gaussian bounds and Collisions of variable speed random walks on lattices with power law conductances

TL;DR

This paper establishes Gaussian bounds for variable speed random walks on weighted lattices with power law conductances and demonstrates infinite collisions of independent walks in specific dimensions and conductance regimes.

Contribution

It provides new Gaussian bounds using an intrinsic metric and analyzes collision behavior for random walks with power law conductances.

Findings

Heat kernel satisfies two-sided Gaussian bounds.

Independent walks collide infinitely often in 2D with specific conductance parameters.

Gaussian bounds derived using an intrinsic metric.

Abstract

We consider a weighted lattice $Z^d$ with conductance $\mu_e=|e|^{-\alpha}$. We show that the heat kernel of a variable speed random walk on it satisfies a two-sided Gaussian bound by using an intrinsic metric. We also show that when $d=2$ and $\alpha\in (-1,0)$, two independent random walks on such weighted lattice will collide infinite many times while they are transient.