The time constant vanishes only on the percolation cone in directed first passage percolation
Yu Zhang, University of Colorado
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 at $vec{p}_c$, as follows:
(1) If $F(0) < vec{p}_c$, then $vec{mu}_F(theta) >0$ for all
$0leq thetaleq pi/2$.
(2) If $F(0) = vec{p}_c$, then $vec{mu}_F(theta) >0$ if and only if $thetaneq pi/4$.
(3) If $F(0)=p > vec{p}_c$, then there exists a percolation cone between $theta_p^-$ and $theta_p^+$ for
$0leq theta^-_p< theta^+_p leq pi/2$ such that $vec{mu} (theta) >0$ if and only if $thetanotin [theta_p^-, theta^+_p]$. Furthermore, all the moments of $vec{T}({bf 0}, (r, theta))$ converge whenever $thetain [theta_p^-, theta^+_p]$.
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 $vec{p}_c$
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