On the Transient (T) condition for Random Walk in Strong Mixing Environment

TL;DR

This paper establishes a law of large numbers and an invariance principle for random walks in strong mixing environments under a weakened condition, using novel renormalization techniques adapted from i.i.d. cases.

Contribution

It weakens the Kalikow ballistic assumption for mixing environments and introduces renormalization schemes for such settings.

Findings

Proves ballistic law of large numbers for random walks in strong mixing environments.

Establishes an invariance principle under the (T) condition.

Demonstrates finite moments of arbitrary order for the regeneration time.

Abstract

We prove a ballistic strong law of large numbers and an invariance principle for random walks in strong mixing environments, under condition $(T)$ of Sznitman (cf. \cite{Sz01}). This weakens by first time the Kalikow ballistic assumption in mixing and proves finite moments of arbitrary order for the approximate regeneration time of \cite{CZ02}. The main technical tool in the proof is the introduction of renormalization schemes, which had only been considered for i.i.d. environments.