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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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