Quenched scaling limit for biased random walks on random, heavy tailed conductances: low dimensions

TL;DR

This paper proves a quenched scaling limit for biased random walks on heavy-tailed conductances in low dimensions, extending previous annealed results to a more robust, almost sure setting.

Contribution

It establishes the quenched scaling limit for biased random walks with heavy-tailed conductances, generalizing prior annealed results to all dimensions.

Findings

Quenched scaling limit matches the fractional kinetics process.

Results hold for all dimensions $d \, \ge \, 2$.

Extends previous annealed convergence to quenched setting.

Abstract

We consider a random walk amongst positive random conductances on $\mathbb{Z}^d, d \ge 2$, with directional bias. When the conductances have a stable distribution with parameter $\gamma \in (0, 1)$, the walk is sub-ballistic. In this regime Fribergh and Kious (Ann. Prob. 2018) derived an annealed scaling limit for the appropriately rescaled walk towards the Fractional Kinetics process. We prove the quenched version of this result for all $d \ge 2$.