Journal of University of Chinese Academy of Sciences >
SLLN and CLT for patterns on the infinite open cluster
Received date: 2013-05-06
Revised date: 2013-07-31
Online published: 2014-07-15
Supported by
Supported by National Natural Science Foundation of China (71271204,11331012,11101420)
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.
TANG Pengfei , GUO Tiande . SLLN and CLT for patterns on the infinite open cluster[J]. Journal of University of Chinese Academy of Sciences, 2014 , 31(4) : 445 -452 . DOI: 10.7523/j.issn.2095-6134.2014.04.001
[1] Kesten H. On the number of self-avoiding walks[J]. Journal of Mathematical Physics, 1963, 4(7): 960-969.
[2] Madras N, Slade G. The self-avoiding walk[M]. Birkhauser, Boston, 1993: 229-242.
[3] Madras N. A pattern theorem for lattice clusters[J]. Annals of Combinatorics, 1999, 3(2): 357-384.
[4] van der Hofstad R, Kager W. Pattern theorems, ratio limit theorems and gumbel maximal clusters for random fields[J]. Journal of Statistical Physics, 2008,130(3): 503-522.
[5] Grimmett G. Percolation[M]. 2nd ed. Berlin: Springer-Verlag, 1999.
[6] Mcleish D L. Dependent central limit theorems and invariance principles[J]. Annals of Probability, 1974, 2(4):620-628.
[7] Kesten H, Zhang Y. A central limit theorem for critical first passage percolation in two dimensions[J]. Probability Theory and Related Fields, 1997, 107(2): 137-160.
[8] Zhang Y. A martingale approach in the study of percolation clusters on the Zd lattice[J]. Journal of Theoretical Probability, 2001, 14(1): 165-187.
[9] Penrose M. A central limit theorem with applications to percolation, epidemics and boolean models[J]. Annals of Probability, 2001, 29(4): 1515-1546.
[10] Jiang J P, Zhang S G, Guo T D. Convergence rate in a martingale CLT for percolation clusters [J]. Journal of the Graduate School of the Chinese Academy of Science, 2010, 27(5): 577-583.
[11] [JP5]Kesten H. Scaling relations for 2D-percolation[J]. Communications in Mathematical Physics, 1987, 109(1):109-156.
[12] Kesten H, Zhang Y. The probability of a large finite cluster in supercritical Bernoulli percolation [J]. Annals of Probability, 1990, 18(2): 537-555.
/
| 〈 |
|
〉 |