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

TL;DR

This paper establishes limit theorems for first-passage percolation on the triangular lattice with Bernoulli weights, providing precise asymptotic behavior and variance, and confirming conjectures related to critical percolation.

Contribution

It proves new almost sure and in-probability limit theorems for passage times, resolving a question by Kesten and Zhang and extending previous results.

Findings

Asymptotic ratio of passage time to log n converges to a constant.

Variance of passage time scaled by log n converges to a specific constant.

Central limit theorem for passage times derived from the asymptotic results.

Abstract

Consider (independent) first-passage percolation on the sites of 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 $b_{0,n}$ the passage time from 0 to the halfplane $\{v\in\mathbb{T}:\mbox{Re}(v)\geq n\}$, and by $T(0,nu)$ the passage time from 0 to the nearest site to $nu$, where $|u|=1$. We prove that as $n\rightarrow\infty$, $b_{0,n}/\log n\rightarrow 1/(2\sqrt{3}\pi)$ a.s., $E[b_{0,n}]/\log n\rightarrow 1/(2\sqrt{3}\pi)$ and Var$[b_{0,n}]/\log n\rightarrow 2/(3\sqrt{3}\pi)-1/(2\pi^2)$; $T(0,nu)/\log n\rightarrow 1/(\sqrt{3}\pi)$ in probability but not a.s., $E[T(0,nu)]/\log n\rightarrow 1/(\sqrt{3}\pi)$ and Var$[T(0,nu)]/\log n\rightarrow 4/(3\sqrt{3}\pi)-1/\pi^2$. 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 CLE$_6$, given by Schramm, Sheffield and Wilson (2009).