Limit theorems for critical first-passage percolation on the triangular lattice

Abstract

Consider (independent) first-passage percolation on the sites of the triangular lattice T embedded in C. 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 b0,n the passage time from 0 to the halfplane {v∈T:Re(v)≥n}, and by T(0,nu) the passage time from 0 to the nearest site to nu, where |u|=1. We prove that as n→∞, b0,n∕logn→1∕(23π) a.s., E[b0,n]∕logn→1∕(23π) and Var[b0,n]∕logn→2∕(33π)−1∕(2π2); T(0,nu)∕logn→1∕(3π) in probability but not a.s., E[T(0,nu)]∕logn→1∕(3π) and Var[T(0,nu)]∕logn→4∕(33π)−1∕π2. This answers a question of Kesten and Zhang (1997) and improves our previous work (2014). From this result, we derive an explicit form of the central limit theorem for b0,n and T(0,nu). A key ingredient for the proof is the moment generating function of the conformal radii for conformal loop ensemble CLE6, given by Schramm et al. (2009).