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

Abstract

We study the site version of (independent) first-passage percolation on the triangular lattice $\mathbb{T}$. Denote the passage time of the site $v$ in $\mathbb{T}$ by $t(v)$, and assume that $P(t(v)=0)=P(t(v)=1)=1/2$. Denote by $a_{0,n}$ the passage time from $\text{0}$ to $(n,0)$, and by $b_{0,n}$ the passage time from $\text{0}$ to the halfplane $\{(x,y):x\ge n\}$. We prove that there exists a constant $0<\mu <\infty$ such that as $n\to \infty$, $a_{0,n}/\log n\to \mu$ in probability and $b_{0,n}/\log n\to \mu /2$ almost surely. This result confirms a prediction of Kesten and Zhang (Probab. Theory Relat. Fields \textbf{107}: 137--160, 1997). The proof relies on the existence of the full scaling limit of critical site percolation on $\mathbb{T}$, established by Camia and Newman.

14 pages, 2 figures

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• Conformal radii for conformal loop ensembles
• On the scaling limits of planar percolation
• On monochromatic arm exponents for 2D critical percolation
• Percolation since Saint-Flour