Critical first-passage percolation starting on the boundary

TL;DR

This paper analyzes first-passage percolation on a 2D triangular lattice starting at the boundary, establishing asymptotic behaviors and limit theorems for passage times with explicit constants.

Contribution

It provides the first rigorous asymptotic results and explicit limit theorems for boundary-starting first-passage percolation on the triangular lattice.

Findings

Limit of normalized passage time c_n^+ / log n equals sqrt(3)/(2π).

Expected passage time E c_n^+ / log n converges to sqrt(3)/(2π).

Variance of c_n^+ / log n converges to (2√3/π) - (9/π^2).

Abstract

We consider first-passage percolation on the two-dimensional triangular lattice $\mathcal{T}$. Each site $v\in\mathcal{T}$ is assigned independently a passage time of either $0$ or $1$ with probability $1/2$. Denote by $B^+(0,n)$ the upper half-disk with radius $n$ centered at $0$, and by $c_n^+$ the first-passage time in $B^+(0,n)$ from $0$ to the half-circular boundary of $B^+(0,n)$. We prove \[\lim_{n\rightarrow\infty}\frac{c_n^+}{\log n}=\frac{\sqrt{3}}{2\pi}~ a.s.,~\lim_{n\rightarrow\infty}\frac{E c_n^+}{\log n}=\frac{\sqrt{3}}{2\pi},~\lim_{n\rightarrow\infty}\frac{\mathrm{Var}(c_n^+)}{\log n}=\frac{2\sqrt{3}}{\pi}-\frac{9}{\pi^2}.\] These results enable us to prove limit theorems with explicit constants for any first-passage time between boundary points of Jordan domains. In particular, we find the explicit limit theorems for the cylinder point to point and cylinder point to line first-passage times.