渗流簇中鞅中心极限定理的收敛速度

姜建平; 张三国; 郭田德

doi:10.7523/j.issn.2095-6134.2010.5.001

中国科学院大学学报 >

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

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

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渗流簇中鞅中心极限定理的收敛速度

姜建平 ,
张三国 ,
郭田德

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中国科学院研究生院数学科学学院, 北京 100049

收稿日期: 2010-03-09

修回日期: 2010-04-16

网络出版日期: 2010-09-15

基金资助

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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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

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摘要

考虑定义在Z^d上参数为p的边渗流模型.假设K_n为 [-n,n]^d中开簇的个数,研究了关于K_n的鞅中心定理的收敛速度.一般情况下,经鞅中心极限定理的最好收敛速度是O(n^-d/2),而我们的结果为P_p((K_n-E_p(K_n))/(Var_p(K_n))) ≤x =^x∫_-∞(1/(2π)) e^(-y²)/2dy+o(n^{-d/2 +ε₀})对任意的实数x都成立,这里ε₀是区间 0, d/2 上的任意实数.据我们所知,这是关于渗流中心极限定理收敛速度的第一结果.

关键词： 渗流; 鞅; 中心极限定理; 收敛速度

本文引用格式

姜建平 , 张三国 , 郭田德 . 渗流簇中鞅中心极限定理的收敛速度[J]. 中国科学院大学学报, 2010 , 27(5) : 577 -583 . DOI: 10.7523/j.issn.2095-6134.2010.5.001

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

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