KMS Of Academy of mathematics and systems sciences, CAS
Limit theorems for critical first-passage percolation on the triangular lattice | |
Yao, Chang-Long | |
2018-02-01 | |
Source Publication | STOCHASTIC PROCESSES AND THEIR APPLICATIONS |
ISSN | 0304-4149 |
Volume | 128Issue:2Pages:445-460 |
Abstract | 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. |
Keyword | 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 |
Language | 英语 |
Funding Project | National Natural Science Foundation of China[11601505] ; Key Laboratory of Random Complex Structures and Data Science, CAS[2008DP173182] |
WOS Research Area | Mathematics |
WOS Subject | Statistics & Probability |
WOS ID | WOS:000423887100004 |
Publisher | ELSEVIER SCIENCE BV |
Citation statistics | |
Document Type | 期刊论文 |
Identifier | http://ir.amss.ac.cn/handle/2S8OKBNM/29470 |
Collection | 应用数学研究所 |
Corresponding Author | Yao, Chang-Long |
Affiliation | Chinese Acad Sci, Acad Math & Syst Sci, Beijing, Peoples R China |
Recommended Citation 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. |
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