文章基本信息

标题：Law of large numbers for critical first-passage percolation on the triangular lattice
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作者：Yao, Chang-Long
期刊名称：Electronic Communications in Probability
印刷版ISSN：1083-589X
出版年度：2014
卷号：19
页码：1-14
DOI：10.1214/ECP.v19-3268
语种：English
出版社：Electronic Communications in Probability
摘要：We study the site version of (independent) first-passage percolation on the triangular lattice $T$. Denote the passage time of the site $v$ in $T$ by $t(v)$, and assume that $\mathbb{P}(t(v)=0)=\mathbb{P}(t(v)=1)=1/2$. Denote by $a_{0,n}$ the passage time from 0 to (n,0), and by b_{0,n} the passage time from 0 to the halfplane $\{(x,y) : x\geq n\}$. We prove that there exists a constant $0<\mu<\infty$ such that as $n\rightarrow\infty$, $a_{0,n}/\log n\rightarrow \mu$ in probability and $b_{0,n}/\log n\rightarrow \mu/2$ almost surely. This result confirms a prediction of Kesten and Zhang. The proof relies on the existence of the full scaling limit of critical site percolation on $T$, established by Camia and Newman.
关键词：critical percolation; first-passage percolation; scaling limit; conformal loop ensemble; law of large numbers;60K35;82B43