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 $\textbf{0}$ to $(n,0)$, and by $b_{0,n}$ the passage time from $\textbf{0}$ 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 (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.