Coupling between Brownian motion and random walks on the infinite percolation cluster
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866911089511366656 |
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| author | Gu, Chenlin Su, Zhonggen Xu, Ruizhe |
| author_facet | Gu, Chenlin Su, Zhonggen Xu, Ruizhe |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_04778 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Coupling between Brownian motion and random walks on the infinite percolation cluster Gu, Chenlin Su, Zhonggen Xu, Ruizhe Probability 35B27, 60K37, 60K35 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. |
| title | Coupling between Brownian motion and random walks on the infinite percolation cluster |
| topic | Probability 35B27, 60K37, 60K35 |
| url | https://arxiv.org/abs/2411.04778 |