Polynomial ballisticity conditions in mixing environments

TL;DR

This paper establishes polynomial ballisticity conditions for random walks in mixing environments, proving ballistic behavior and a central limit theorem, and clarifies the equivalence of certain conditions in this setting.

Contribution

It introduces a new mixing effective criterion implied by polynomial conditions, proving key conjectures about the equivalence of ballisticity conditions in mixing environments.

Findings

Proves ballistic behavior under polynomial conditions in mixing environments.

Establishes an annealed functional central limit theorem for RWRE.

Clarifies the equivalence of conditions $(T^eta)| ext{ extlangle} angle$ in mixing environments.

Abstract

We prove ballistic behaviour as well as an annealed functional central limit theorem for random walks in mixing random environments (RWRE). The ballistic hypothesis will be an effective polynomial condition as the one introduced by Berger, Drewitz, and Ram\'{\i}rez (\emph{Comm. Pure Appl. Math,} {\bf 67}, (2014) 1947--1973). The novel idea therein was the construction of several simultaneous renormalization steps, providing more flexibility for seed estimates. For our proof, we indeed follow a similar path, and introduce a new mixing effective criterion which will be implied by the polynomial condition. This allows us to prove, in a mixing framework, the RWRE conjecture concerning the equivalence between each condition $(T^\gamma)|\ell$, for $\gamma\in (0,1)$ and $\ell \in \mathbb S^{d-1}$. This work complements the previous work of Guerra (\emph{Ann. Probab.} {\bf 47} (2019) 3003--3054) and completes the answer about the meaning of condition $(T')|\ell$ in a mixing setting, an open question posed by Comets and Zeitouni (\emph{Ann. Probab.} {\bf 32} (2004) 880--914).