Limit theorems for critical first-passage percolation on the triangular lattice

doi:10.1016/j.spa.2017.05.002

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	Limit theorems for critical first-passage percolation on the triangular lattice
	Yao, Chang-Long
	2018-02-01
发表期刊	STOCHASTIC PROCESSES AND THEIR APPLICATIONS
ISSN	0304-4149
卷号	128 期号:2 页码:445-460
摘要	Consider (independent) first-passage percolation on the sites of the triangular lattice T embedded in C. Denote the passage time of the site v in T by t(v), and assume that P(t(v) = 0) = P(t(v) = 1) = 1/2. Denote by b(0,n) the passage time from 0 to the halfplane {v is an element of T : Re(v) >= n}, and by T(0, nu) the passage time from 0 to the nearest site to nu, where vertical bar u vertical bar = 1. We prove that as n -> infinity, b(0,n)/ log n -> 1/(2 root 3 pi) a.s., E[b(0,n)]/ log n -> 1/(2 root 3 pi) and Var[b(0,n)]/ log -> n 2/(3 root 3 pi) - 1/(2 pi(2)); T(0, nu)/ log n -> 1/(root 3 pi) in probability but not a.s., E[T (0, nu)]/ log n -> 1/(root 3 pi) and Var[T(0, nu)]/ log n -> 4/(3 root 3 pi) - 1/pi(2). This answers a question of Kesten and Zhang (1997) and improves our previous work (2014). From this result, we derive an explicit form of the central limit theorem for b(0,n) and T (0, nu). A key ingredient for the proof is the moment generating function of the conformal radii for conformal loop ensemble CLE6, given by Schramm et al. (2009). (C) 2017 Elsevier B.V. All rights reserved.
关键词	Critical percolation First-passage percolation Scaling limit Conformal loop ensemble Law of large numbers Central limit theorem
DOI	10.1016/j.spa.2017.05.002
语种	英语
资助项目	National Natural Science Foundation of China[11601505] ; Key Laboratory of Random Complex Structures and Data Science, CAS[2008DP173182]
WOS研究方向	Mathematics
WOS类目	Statistics & Probability
WOS记录号	WOS:000423887100004
出版者	ELSEVIER SCIENCE BV
引用统计
文献类型	期刊论文
条目标识符	http://ir.amss.ac.cn/handle/2S8OKBNM/29470
专题	应用数学研究所
通讯作者	Yao, Chang-Long
作者单位	Chinese Acad Sci, Acad Math & Syst Sci, Beijing, Peoples R China
推荐引用方式 GB/T 7714	Yao, Chang-Long. Limit theorems for critical first-passage percolation on the triangular lattice[J]. STOCHASTIC PROCESSES AND THEIR APPLICATIONS,2018,128(2):445-460.
APA	Yao, Chang-Long.(2018).Limit theorems for critical first-passage percolation on the triangular lattice.STOCHASTIC PROCESSES AND THEIR APPLICATIONS,128(2),445-460.
MLA	Yao, Chang-Long."Limit theorems for critical first-passage percolation on the triangular lattice".STOCHASTIC PROCESSES AND THEIR APPLICATIONS 128.2(2018):445-460.