The Time Constant Vanishes Only on the Percolation Cone in Directed First Passage Percolation

Abstract

We consider the directed first passage percolation model on $\mathbb{Z}^2$. In this model, we assign independently to each edge $e$ a passage time $t(e)$ with a common distribution $F$. We denote by $\vec{T}(0,(r,\theta))$ the passage time from the origin to $(r,\theta)$ by a northeast path for $(r,\theta)\in\mathbb{R}_+\times[0,\pi/2]$. It is known that $\vec{T}(0,(r,\theta))/r$ 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 at $\vec{p}_c$, 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 $\vec{T}(0, (r, \theta))$ 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 at $\vec{p}_c$