A conditional quenched CLT for random walks among random conductances on $\mathbb{Z}^d$

TL;DR

This paper establishes a conditional quenched central limit theorem for random walks in random conductances on integer lattices, showing the limit law as a product of a Brownian meander and a Brownian motion.

Contribution

It introduces a new conditional limit law for random walks among random conductances, extending quenched CLT results to conditioned paths.

Findings

The limit law is a product of a Brownian meander and a (d-1)-dimensional Brownian motion.

The result applies to random walks conditioned to have their first coordinate positive.

The study advances understanding of conditioned diffusive behavior in random environments.

Abstract

Consider a random walk among random conductances on $\mathbb{Z}^d$ with $d\geq 2$. We study the quenched limit law under the usual diffusive scaling of the random walk conditioned to have its first coordinate positive. We show that the conditional limit law is the product of a Brownian meander and a $(d-1)$-dimensional Brownian motion.