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This is a Quantitative Study study.

October 8, 2025Open Access

Quenched limit for diffusive biased random walks in random environment

Key Points

Every directionally transient random walk shows a quenched limit in multiple dimensions.
Under condition (T)γ, these random walks exhibit a functional limit towards Brownian motion.
Almost-optimal variance bounds on functionals of the random walk's quenched expectation are achieved.
The classical method used has roots in the foundational work by Bolthausen and Sznitman.

Abstract

We prove that every directionally transient random walk in random i. i. d. \ environment, under condition (T) _, which admits an annealed functional limit towards Brownian motion also admits the corresponding quenched limit in d 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.

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Cite This Study

Carlo Scali (Fri,) studied this question.

synapsesocial.com/papers/68e5c1c76950a706b22b5e2c https://doi.org/https://doi.org/10.48550/arxiv.2506.11799

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