Random walks with unbounded jumps among random conductances I: Uniform quenched CLT

TL;DR

This paper proves a uniform quenched invariance principle for one-dimensional random walks with unbounded jumps in random conductances, showing convergence to Brownian motion uniformly over a range of starting points.

Contribution

It establishes a uniform invariance principle for random walks with unbounded jumps under ergodic conductances, extending previous results to a broader setting.

Findings

Proves a quenched uniform invariance principle for the walk.

Shows convergence to Brownian motion uniformly over starting locations.

Handles unbounded jumps with polynomial tail bounds.

Abstract

We study a one-dimensional random walk among random conductances, with unbounded jumps. Assuming the ergodicity of the collection of conductances and a few other technical conditions (uniform ellipticity and polynomial bounds on the tails of the jumps) we prove a quenched \textit{uniform} invariance principle for the random walk. This means that the rescaled trajectory of length $n$ is (in a certain sense) close enough to the Brownian motion, uniformly with respect to the choice of the starting location in an interval of length $O(\sqrt{n})$ around the origin.