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Limit theorems for critical first-passage percolation on the triangular lattice
Yao, Chang-Long
AbstractConsider (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.
KeywordCritical percolation First-passage percolation Scaling limit Conformal loop ensemble Law of large numbers Central limit theorem
Funding ProjectNational Natural Science Foundation of China[11601505] ; Key Laboratory of Random Complex Structures and Data Science, CAS[2008DP173182]
WOS Research AreaMathematics
WOS SubjectStatistics & Probability
WOS IDWOS:000423887100004
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Document Type期刊论文
Corresponding AuthorYao, Chang-Long
AffiliationChinese 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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