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 Electronic Journal of Probability > Vol. 14 (2009) > Paper 77 open journal systems 


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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Pages: 2264-2286

Published on: October 16, 2009
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Electronic Journal of Probability. ISSN: 1083-6489