Convergence rate in a martingale CLT for percolation clusters

JIANG Jian-Ping; ZHANG San-Guo; GUO Tian-De

doi:10.7523/j.issn.2095-6134.2010.5.001

Journal of University of Chinese Academy of Sciences >

2010 , Vol. 27 >Issue 5: 577 - 583

DOI: https://doi.org/10.7523/j.issn.2095-6134.2010.5.001

Research Articles

Convergence rate in a martingale CLT for percolation clusters

JIANG Jian-Ping ,
ZHANG San-Guo ,
GUO Tian-De

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School of Mathematical Sciences, Graduate University, Chinese Academy of Sciences, Beijing 100049, China

Received date: 2010-03-09

Revised date: 2010-04-16

Online published: 2010-09-15

Supported by

Supported by Knowledge Innovation Program of the Chinese Academy of Sciences(kjcx-yw-s7), the National Natural Science Foundation of China (10831006) and Presidential Foundation of GUCAS(O85101BM03) 

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Abstract

Consider bond percolation on Z^d with parameter p. Let K_n be the number of open clusters in [-n,n]^d. We investigate the convergence rate in the martingaleCLT for K_n. Generally, the best convergence rate for classicalmartingale CLT is O(n^-d/2), and our result is P_p((K_n-E_p(K_n))/(Var_p(K_n))) ≤x =^x∫_-∞(1/(2π)) e^(-y²)/2dy+o(n^{-d/2 +ε₀}) for all x, where ε₀ is any constant real number in 0, d/2 . As far as we know, this is the first convergence rate in CLTs for percolation.

Key words： percolation; martingale; central limit theorem; rate of convergence

Cite this article

JIANG Jian-Ping , ZHANG San-Guo , GUO Tian-De . Convergence rate in a martingale CLT for percolation clusters[J]. Journal of University of Chinese Academy of Sciences, 2010 , 27(5) : 577 -583 . DOI: 10.7523/j.issn.2095-6134.2010.5.001

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