Quenched limit for diffusive biased random walks in random environment

TL;DR

This paper establishes that in higher dimensions, certain biased random walks in random environments exhibit a quenched limit behavior similar to their annealed limit, under specific conditions, with improved variance bounds.

Contribution

It proves the quenched invariance principle for biased random walks in i.i.d. environments under condition (T)_{ ext{γ}}, with near-optimal variance bounds, extending previous results.

Findings

Quenched invariance principle established in dimensions d ≥ 2.

Almost-optimal bounds on variance of quenched expectations.

Extension of classical strategies to improve variance estimates.

Abstract

We prove that every directionally transient random walk in random i.i.d.\ environment, under condition $(T)_{\gamma}$, which admits an annealed functional limit towards Brownian motion also admits the corresponding quenched limit in $d \ge 2$. We exploit a classical strategy that was introduced by Bolthausen and Sznitman but, with respect to the existing literature, we get almost-optimal bounds on the variance of the quenched expectation of certain functionals of the random walk.