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2014 Law of large numbers for critical first-passage percolation on the triangular lattice
Chang-Long Yao
Author Affiliations +
Electron. Commun. Probab. 19: 1-14 (2014). DOI: 10.1214/ECP.v19-3268

Abstract

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.

Citation

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Chang-Long Yao. "Law of large numbers for critical first-passage percolation on the triangular lattice." Electron. Commun. Probab. 19 1 - 14, 2014. https://doi.org/10.1214/ECP.v19-3268

Information

Accepted: 15 March 2014; Published: 2014
First available in Project Euclid: 7 June 2016

zbMATH: 1315.60114
MathSciNet: MR3183571
Digital Object Identifier: 10.1214/ECP.v19-3268

Subjects:
Primary: 60K35
Secondary: 82B43

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