Electronic Communications in Probability

Law of large numbers for critical first-passage percolation on the triangular lattice

Chang-Long Yao

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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.

Article information

Electron. Commun. Probab., Volume 19 (2014), paper no. 18, 14 pp.

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

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Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82B43: Percolation [See also 60K35]

critical percolation first-passage percolation scaling limit conformal loop ensemble law of large numbers

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Yao, Chang-Long. Law of large numbers for critical first-passage percolation on the triangular lattice. Electron. Commun. Probab. 19 (2014), paper no. 18, 14 pp. doi:10.1214/ECP.v19-3268. https://projecteuclid.org/euclid.ecp/1465316720

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