






















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.
此内容由惯性聚合(RSS阅读器)自动聚合整理,仅供阅读参考。 原文来自 — 版权归原作者所有。