Electronic Communications in Probability

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

Chang-Long Yao

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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.

Article information

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465316720

Digital Object Identifier
doi:10.1214/ECP.v19-3268

Mathematical Reviews number (MathSciNet)
MR3183571

Zentralblatt MATH identifier
1315.60114

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

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

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

Citation

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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