A proof of Sznitman's conjecture about ballistic RWRE

TL;DR

This paper proves Sznitman's conjecture that certain ballisticity conditions in random walks in random environments are equivalent, establishing a key link between different criteria for ballistic behavior in high-dimensional settings.

Contribution

It demonstrates the equivalence of conditions (T), (T'), and (T)_γ, confirming a long-standing conjecture and connecting these with the polynomial condition (P)_M.

Findings

Proves the equivalence of conditions (T) and (T') in 2002 conjecture.

Establishes the equivalence of (T), (T') and (T)_γ for some γ in (0,1).

Links ballisticity conditions with the polynomial condition (P)_M for M ≥ 15d+5.

Abstract

We consider a random walk in a uniformly elliptic i.i.d. random environment in $\mathbb Z^d$ for $d\ge 2$. It is believed that whenever the random walk is transient in a given direction it is necessarily ballistic. In order to quantify the gap which would be needed to prove this equivalence, several ballisticity conditions have been introduced. In particular, in 2001 and 2002, Sznitman defined the so called conditions $(T)$ and $(T')$. The first one is the requirement that certain unlikely exit probabilities from a set of slabs decay exponentially fast with their width $L$. The second one is the requirement that for all $\gamma\in (0,1)$ condition $(T)_\gamma$ is satisfied, which in turn is defined as the requirement that the decay is like $e^{-CL^\gamma}$ for some $C>0$. In this article we prove a conjecture of Sznitman of 2002, stating that $(T)$ and $(T')$ are equivalent. Hence, this closes the circle proving the equivalence of conditions $(T)$, $(T')$ and $(T)_\gamma$ for some $\gamma\in (0,1)$ as conjectured by Sznitman, and also of each of these ballisticity conditions with the polynomial condition $(P)_M$ for $M\ge 15d+5$ introduced by Berger, Drewitz and Ramirez in 2014.