Subdiffusivity of random walk on the 2D invasion percolation cluster

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• 作者：Michael Damron ; Jack Hanson ; Philippe Sosoe
• 关键词：Percolation ; Invasion ; Incipient infinite cluster ; Subdiffusivity ; Criticality
• 刊名：Stochastic Processes and their Applications
• 出版年：2013
• 出版时间：September, 2013
• 年：2013
• 卷：123
• 期：9
• 页码：3588-3621
• 全文大小：350 K

文摘

We derive quenched subdiffusive lower bounds for the exit time from a box of size for the simple random walk on the planar invasion percolation cluster. The first part of the paper is devoted to proving an almost sure analogue of H. Kesten¡¯s subdiffusivity theorem for the random walk on the incipient infinite cluster and the invasion percolation cluster using ideas of M. Aizenman, A. Burchard and A. Pisztora. The proof combines lower bounds on the intrinsic distance in these graphs and general inequalities for reversible Markov chains. In the second part of the paper, we present a sharpening of Kesten¡¯s original argument, leading to an explicit almost sure lower bound for in terms of percolation arm exponents. The methods give , where depends on the intrinsic distance and can be taken to be on the hexagonal lattice.