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

Yu Zhang
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Yu Zhang: University of Colorado

Journal of Theoretical Probability, 2001, vol. 14, issue 1, 165-187

Abstract: 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.

Keywords: Martingale; percolation; large deviation (search for similar items in EconPapers)
Date: 2001
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DOI: 10.1023/A:1007877216583

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