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Coupling between Brownian motion and random walks on the infinite percolation cluster
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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.
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The diffusivity of supercritical Bernoulli percolation is infinitely differentiable
The mapping p to sigma(p) for supercritical bond percolation on Z^d is C^infinity on (p_c,1], a full-interval extension of Kozlov's 1989 result.
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