Conditions for ballisticity and invariance principle for random walk in non-elliptic random environment

TL;DR

This paper establishes conditions under which random walks in non-elliptic i.i.d. environments on integer lattices exhibit ballistic behavior and satisfy an invariance principle, extending classical results beyond elliptic cases.

Contribution

It introduces novel local conditions for ballisticity and an annealed invariance principle applicable to non-elliptic random environments, using oriented percolation and martingale techniques.

Findings

Derived non-trivial local conditions for ballisticity.

Proved an annealed invariance principle in non-elliptic settings.

Extended classical results to non-elliptic environments.

Abstract

We study the asymptotic behaviour of random walks in i.i.d. non-elliptic random environments on $\mathbb{Z}^d$. Standard conditions (and proofs) for ballisticity and the central limit theorem require ellipticity. We use oriented percolation and martingale arguments to give non-trivial local conditions for ballisticity and an annealed invariance principle in the non-elliptic setting.