Quenched invariance principle for biased random walks in random conductances in the sub-ballistic regime

TL;DR

This paper proves that in high dimensions, a biased random walk in random conductances, when properly rescaled in the sub-ballistic regime, converges to a Fractional Kinetics process, revealing its long-term behavior.

Contribution

It establishes the quenched invariance principle for biased random walks in random conductances in the sub-ballistic regime, a significant extension in high-dimensional settings.

Findings

Quenched convergence to Fractional Kinetics in high dimensions

Validates the sub-ballistic regime behavior of biased random walks

Extends invariance principles to complex random environments

Abstract

We consider a biased random walk in positive random conductances on $\mathbb{Z}^d$ for $d\geq 5$. In the sub-ballistic regime, we prove the quenched convergence of the properly rescaled random walk towards a Fractional Kinetics.