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

2016/01/30 by Yao, Chang-Long
#60K35 #82B43 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1602.00065

Abstract

Consider (independent) first-passage percolation on the sites of the triangular lattice \mathbbT. Denote the passage time of the site v in \mathbbT by t(v), and assume that P(t(v)=0)=P(t(v)=1)=1/2. Denote by b_0,n the passage time from 0 to the halfplane \v∈\mathbbT: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→∞, b_0,n/log n→ 1/(2√(3)π) a.s., E[b_0,n]/log n→ 1/(2√(3)π) and Var[b_0,n]/log n→ 2/(3√(3)π)-1/(2π²); T(0,nu)/log n→ 1/(√(3)π) in probability but not a.s., E[T(0,nu)]/log n→ 1/(√(3)π) and Var[T(0,nu)]/log n→ 4/(3√(3)π)-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 b_0,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, Sheffield and Wilson (2009).

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