An invariance principle for one-dimensional random walks in degenerate dynamical random environments

TL;DR

This paper proves that one-dimensional random walks in certain degenerate, time-dependent random environments converge to Brownian motion under diffusive scaling, despite the conductances potentially vanishing over time.

Contribution

It establishes an invariance principle for random walks in degenerate dynamical environments with minimal assumptions, extending previous results to more general settings.

Findings

Random walks scale to non-degenerate Brownian motion.

Results apply to dynamical percolation with general edge-flip dynamics.

Convergence holds for almost every environment realization.

Abstract

We study random walks on the integers driven by a sample of time-dependent nearest-neighbor conductances that are bounded but are permitted to vanish over time intervals of positive Lebesgue-length. Assuming only ergodicity of the conductance law under space-time shifts and a moment assumption on the time to accumulate a unit conductance over a given edge, we prove that the walk scales, under a diffusive scaling of space and time, to a non-degenerate Brownian motion for a.e. realization of the environment. The conclusion particularly applies to random walks on one-dimensional dynamical percolation subject to fairly general stationary edge-flip dynamics.