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

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 P (t(v) = 0) = 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 >= n}. We prove that there exists a constant 0 < mu < infinity such that as n -> infinity, a(0,n)/log n -> mu in probability and b(0,n)/log n -> 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.