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

2024/11/07 by Chenlin Gu, Zhonggen Su, Gu, Chenlin +3
Mathematics · Physics and Astronomy · #35B27 #60K35 #60K37 #FOS: Mathematics #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics #Theoretical and Computational Physics

paper · pdf · doi:10.48550/arxiv.2411.04778

openalex publication_date 2024/11/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For the supercritical Bernoulli bond percolation on ℤ^d (d ≥ 2), 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 T^(1)/(3)+o(1). 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.

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