The time constant vanishes only on the percolation cone in directed first passage percolation

TL;DR

This paper studies the phase transition of the time constant in directed first passage percolation on Z^2, showing it vanishes only within a specific percolation cone related to the critical probability.

Contribution

It characterizes the phase transition of the time constant in directed first passage percolation based on the distribution's mass at zero and the critical percolation probability, identifying a percolation cone where the constant vanishes.

Findings

Time constant positive outside the percolation cone.

Vanishing time constant only occurs within the percolation cone.

Describes shape of the growth model and phase transition at critical probability.

Abstract

We consider the directed first passage percolation model on ${\bf Z}^2$. In this model, we assign independently to each edge $e$ a passage time $t(e)$ with a common distribution $F$. We denote by $\vec{T}({\bf 0}, (r,\theta))$ the passage time from the origin to $(r, \theta)$ by a northeast path for $(r, \theta)\in {\bf R}^+\times [0,\pi/2]$. It is known that $\vec{T}({\bf 0}, (r, \theta))/r$ converges to a time constant $\vec{\mu}_F (\theta)$. Let $\vec{p}_c$ denote the critical probability for oriented percolation. In this paper, we show that the time constant has a phase transition divided by $\vec{p}_c$, as follows: (1) If $F(0) < \vec{p}_c$, then $\vec{\mu}_F(\theta) >0$ for all $0\leq \theta\leq \pi/2$. (2) If $F(0) = \vec{p}_c$, then $\vec{\mu}_F(\theta) >0$ if and only if $\theta\neq \pi/4$. (3) If $F(0)=p > \vec{p}_c$, then there exists a percolation cone between $\theta_p^-$ and $\theta_p^+$ for $0\leq \theta^-_p< \theta^+_p \leq \pi/2$ such that $\vec{\mu} (\theta) >0$ if and only if $\theta\not\in [\theta_p^-, \theta^+_p]$. Furthermore, all the moments of $\vec{T}({\bf 0}, (r, \theta))$ converge whenever $\theta\in [\theta_p^-, \theta^+_p]$. As applications, we describe the shape of the directed growth model on the distribution of $F$. We give a phase transition for the shape divided by $\vec{p}_c$.