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Journal of University of Chinese Academy of Sciences ›› 2014, Vol. 31 ›› Issue (4): 445-452.DOI: 10.7523/j.issn.2095-6134.2014.04.001

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SLLN and CLT for patterns on the infinite open cluster

TANG Pengfei, GUO Tiande   

  1. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
  • Received:2013-05-06 Revised:2013-07-31 Online:2014-07-15
  • Supported by:

    Supported by National Natural Science Foundation of China (71271204,11331012,11101420)

Abstract:

In this paper, we consider supercritical Bernoulli bond percolation on the integer lattice Ld with parameter p. We study occurrences of patterns on the infinite open cluster. Let Λn denote the number of occurrences of a given pattern P on the infinite open cluster restricted in the box B(n)=[-n,n]d. Strong law of large numbers and central limit theorem for Λn are obtained.

Key words: percolation, pattern theorem, martingale, strong law of large numbers, central limit theorem

CLC Number: