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

TL;DR

This paper constructs a coupling between the random walk on the infinite percolation cluster and Brownian motion, revealing a subdiffusive maximum deviation of order T^{1/3+o(1)} and addressing an open problem in percolation theory.

Contribution

It introduces a novel coupling method using optimal transport to analyze the random walk and Brownian motion on the infinite cluster, providing new insights into their relationship.

Findings

Maximum deviation between coupled paths is of order T^{1/3+o(1)}.

Recovers and refines the law of the iterated logarithm for the random walk.

Partially answers an open question by Biskup regarding coupling behavior.

Abstract

For the supercritical Bernoulli bond percolation on $\mathbb{Z}^d$ ($d \geq 2$), 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 $T^{\frac{1}{3}+o(1)}$. 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.