Scaling limits for sub-ballistic biased random walks in random conductances

TL;DR

This paper investigates the scaling behavior of biased random walks in random conductances on lattices, establishing laws of large numbers and central limit theorems that reveal complex limiting processes.

Contribution

It provides the first functional limit theorems for biased random walks in random conductances in the zero-speed regime, including a fractional kinetics process in the limit.

Findings

Proves a functional Law of Large Numbers for the walk's position.

Establishes a functional Central Limit Theorem involving fractional kinetics.

Identifies the limiting process as related to fractional kinetics in the zero-speed regime.

Abstract

We consider biased random walks in positive random conductances on the d-dimensional lattice in the zero-speed regime and study their scaling limits. We obtain a functional Law of Large Numbers for the position of the walker, properly rescaled. Moreover, we state a functional Central Limit Theorem where an atypical process, related to the Fractional Kinetics, appears in the limit.