The random conductance model with Cauchy tails

TL;DR

This paper studies a random walk in an environment with Cauchy-tailed conductances, proving a quenched functional CLT and improving the local limit theorem for uniform convergence of transition probabilities.

Contribution

It introduces a quenched CLT for the walk and enhances the local limit theorem to achieve uniform convergence across a region.

Findings

Established a quenched functional CLT for the model.

Improved the local limit theorem for uniform convergence.

Provided new bounds for transition probabilities in the environment.

Abstract

We consider a random walk in an i.i.d. Cauchy-tailed conductances environment. We obtain a quenched functional CLT for the suitably rescaled random walk, and, as a key step in the arguments, we improve the local limit theorem for $p^{\omega}_{n^2t}(0,y)$ in [Ann. Probab. (2009). To appear], Theorem 5.14, to a result which gives uniform convergence for $p^{\omega}_{n^2t}(x,y)$ for all $x,y$ in a ball.