Ellipticity criteria for ballistic behavior of random walks in random environment

TL;DR

This paper establishes new ellipticity criteria for random walks in i.i.d. environments, extending existing ballisticity conditions and linking them to central limit theorems.

Contribution

It introduces ellipticity criteria that generalize previous conditions, proving their equivalence and implications for ballisticity and limit theorems in random walks.

Findings

Proves equivalence of Sznitman's (T') condition with polynomial effective criteria.

Provides ellipticity conditions ensuring ballisticity of the walk.

Establishes criteria under which the walk satisfies CLT results.

Abstract

We introduce ellipticity criteria for random walks in i.i.d. random environments under which we can extend the ballisticity conditions of Sznitman's and the polynomial effective criteria of Berger, Drewitz and Ramirez originally defined for uniformly elliptic random walks. We prove under them the equivalence of Sznitman's (T') condition with the polynomial effective criterion (P)_M, for M large enough. We furthermore give ellipticity criteria under which a random walk satisfying the polynomial effective criterion, is ballistic, satisfies the annealed central limit theorem or the quenched central limit theorem.