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The time constant vanishes only on the percolation cone in directed first passage percolation. (English) Zbl 1192.60109
Summary: 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 $$\vec{T}({\mathbf 0}, (r,\theta))$$ 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 $$\vec{T}({\mathbf 0}, (r,\theta))/r$$ 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 $$\vec{T}({\mathbf 0}, (r, \theta))$$ 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 $$\vec{p}_c$$.

##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory
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