Homogenization theory of random walks among deterministic conductances

• 成果类型: Article
• 署名作者: Biskup, Marek
• 署名单位: University of California System; University of California Los Angeles
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051; 1432-2064
• DOI: 10.1007/s00440-025-01442-0
• 发表日期: 2026-02
• 页码: 41-102
• 关键词: quenched invariance-principle; central-limit-theorem; REVERSIBLE RANDOM-WALKS; Scaling Limit; Stochastic Homogenization; WEAK-CONVERGENCE; upper-bounds; percolation; corrector; MODEL

摘要

We study the asymptotic distribution of random walks on Zd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb Z<^>d$$\end{document} (d >= 1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d\geqslant 1$$\end{document}) in deterministic reversible environments defined by an assignment of a positive conductance to each edge of Zd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb Z<^>d$$\end{document}. We identify a deterministic set of conductance configurations for which the walk obeys an Invariance Principle; i.e., converges in law to a non-degenerate Brownian motion under diffusive scaling of space and time. This set is closed under translations and zero-density perturbations and carries all ergodic conductance laws subject to certain moment conditions. The proofs rely on martingale approximations whose main step is the conversion of averages in time and physical space under the deterministic environment to those in a suitable stochastic counterpart. Our study sets up a framework for de-randomized homogenization of other motions in disordered media.