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A Martingale Approach in the Study of Percolation Clusters on the Z d Lattice

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Abstract

Consider percolation on Z d with parameter p. Let K n be the number of occupied clusters in [−n, n]d. Here we use a martingale method to show that if p≠0, 1, K n satisfies the CLT for all d>1. Furthermore, we investigate the large deviations and concentration property for K n . Besides K n , we also consider the distribution of the number Λ n of such vertices connected by the infinite occupied cluster in a large box [−n, n]d. We show that Λ n satisfies the CLT and investigate the concentration property for Λ n , by using the martingale method in the supercritical phase.

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Authors and Affiliations

  1. Department of Mathematics, University of Colorado, Colorado Springs, Colorado, 80933

    Yu Zhang

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  1. Yu Zhang
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Zhang, Y. A Martingale Approach in the Study of Percolation Clusters on the Z d Lattice. Journal of Theoretical Probability 14, 165–187 (2001). https://doi.org/10.1023/A:1007877216583

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  • DOI: https://doi.org/10.1023/A:1007877216583

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