Coupling between Brownian motion and random walks on the infinite percolation cluster

• 成果类型: Article; Early Access
• 署名作者: Gu, Chenlin; Su, Zhonggen; Xu, Ruizhe
• 署名单位: Tsinghua University; Zhejiang University
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-025-01403-7
• 发表日期: 2025
• 关键词: quenched invariance-principles; strong approximation theorems; random conductance model; Stochastic Homogenization; large deviations; quantitative homogenization; spectral gap; REGULARITY; CONVERGENCE; uniqueness

摘要

For the supercritical Bernoulli bond percolation on \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} (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d \geqslant 2$$\end{document}), we give a coupling between the random walk on the infinite cluster and its limit Brownian motion, such that the maximum distance between the paths during [0, T] has a mean of order \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T<^>{\frac{1}{3}+o(1)}$$\end{document}. The construction of the coupling utilizes the optimal transport tool. The analysis mainly relies on local CLT and the concentration of the cluster density. This partially answers an open question posed by Biskup (Probab Surv 8:294-373, 2011). As a direct application, our result recovers the law of the iterated logarithm proved by Duminil-Copin (arXiv:0809.4380), and further identifies the limit constant.