The time constant vanishes only on the percolation cone in directed first passage percolation
Abstract
Description
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 $\vec{T}({\bf 0}, (r,θ))$ the passage time from the origin to $(r, θ)$ by a northeast path for $(r, θ)\in {\bf R}^+\times [0,π/2]$. It is known that $\vec{T}({\bf 0}, (r, θ))/r$ converges to a time constant $\vecμ_F (θ)$. 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 $\vecμ_F(θ) >0$ for all $0\leq θ\leq π/2$.
(2) If $F(0) = \vec{p}_c$, then $\vecμ_F(θ) >0$ if and only if $θ\neq π/4$.
(3) If $F(0)=p > \vec{p}_c$, then there exists a percolation cone between $θ_p^-$ and $θ_p^+$ for
$0\leq θ^-_p< θ^+_p \leq π/2$ such that $\vecμ (θ) >0$ if and only if $θ\not\in [θ_p^-, θ^+_p]$. Furthermore, all the moments of $\vec{T}({\bf 0}, (r, θ))$ converge whenever $θ\in [θ_p^-, θ^+_p]$. 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 $\vec{p}_c$.
28 pages, 1 figure
28 pages, 1 figure