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arXiv 0802.3519math.PR
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在定向首程渗流中，时间常数只在渗流锥上为零

The time constant vanishes only on the percolation cone in directed first passage percolation

Yu Zhang

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28 pages, 1 figure
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摘要：

在${\bf Z}^2$上考虑了有向第一通道渗流模型。在这个模型中，我们独立地分配给每个边缘$e$一个通行时间$t(e)$与一个共同的分布$F$。我们用$\vec{T}({\bf 0}, (r,θ))$表示通过$(r, θ)\in {\bf R}^+\times [0,π/2]$的东北路径从原点到$(r, θ)$的通过时间。已知$\vec{T}({\bf 0}, (r, θ))/r$收敛于时间常数$\vecμ_F (θ)$。令$\vec{p}_c$表示定向渗流的临界概率。本文证明了时间常数有一个除以$\vec{p}_c$的相变，如下：（1）若$F(0) < \vec{p}_c$，则对所有的$0\leq θ\leq π/2$都有$\vecμ_F(θ) >0$。(2)如果是$F(0) = \vec{p}_c$，则$\vecμ_F(θ) >0$当且仅当$θ\neq π/4$。(3)如果$F(0)=p > \vec{p}_c$，则对于$0\leq θ^-_p< θ^+_p \leq π/2$，在$θ_p^-$和$θ_p^+$之间存在渗流锥，使得$\vecμ (θ) >0$当且仅当$θ\not\in [θ_p^-, θ^+_p]$。此外，当$θ\in [θ_p^-, θ^+_p]$时，$\vec{T}({\bf 0}, (r, θ))$的所有矩收敛。作为应用，我们描述了$F$分布的定向生长模型的形状。我们给出了一个相变的形状除以$\vec{p}_c$。

英文摘要：

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 $\vec{p}_c$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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