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<p>For the supercritical Bernoulli bond percolation on <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb {Z}^d\)</EquationSource> </InlineEquation> (<InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(d \geqslant 2\)</EquationSource> </InlineEquation>), 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,&#xa0;<i>T</i>] has a mean of order <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(T^{\frac{1}{3}+o(1)}\)</EquationSource> </InlineEquation>. 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 (<a href="http://arxiv.org/abs/0809.4380">arXiv:0809.4380</a>), and further identifies the limit constant.</p>

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

  • 期刊论文

  • 发表日期：

    2025年8月4日

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  • Chenlin Gu,
  • Zhonggen Su,
  • Ruizhe Xu

摘要

For the supercritical Bernoulli bond percolation on \(\mathbb {Z}^d\) ( \(d \geqslant 2\) ), 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.