@article{ECP3268,
author = {Chang-Long Yao},
title = {Law of large numbers for critical first-passage percolation on the triangular lattice},
journal = {Electron. Commun. Probab.},
fjournal = {Electronic Communications in Probability},
volume = {19},
year = {2014},
keywords = {critical percolation; first-passage percolation; scaling limit; conformal loop ensemble; law of large numbers},
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.},
pages = {no. 18, 1-14},
issn = {1083-589X},
doi = {10.1214/ECP.v19-3268},
url = {http://ecp.ejpecp.org/article/view/3268}}