Almost sure functional central limit theorem for ballistic random walk in random environment

TL;DR

This paper proves an almost sure functional central limit theorem for multidimensional ballistic random walks in random environments, showing the walk's scaled behavior converges to a Brownian motion under broad conditions.

Contribution

It establishes an invariance principle for the walk under almost every environment, highlighting the subdiffusive behavior of the quenched mean.

Findings

Invariance principle holds for high moments of regeneration times.

Walk exhibits diffusive scaling behavior in random environments.

Results apply to multidimensional settings with bounded steps.

Abstract

We consider a multidimensional random walk in a product random environment with bounded steps, transience in some spatial direction, and high enough moments on the regeneration time. We prove an invariance principle, or functional central limit theorem, under almost every environment for the diffusively scaled centered walk. The main point behind the invariance principle is that the quenched mean of the walk behaves subdiffusively.