The Time Constant Vanishes Only on the Percolation Cone in Directed First Passage Percolation
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| 1. | Title | Title of document | The Time Constant Vanishes Only on the Percolation Cone in Directed First Passage Percolation |
| 2. | Creator | Author's name, affiliation, country | Yu Zhang; University of Colorado; United States |
| 3. | Subject | Discipline(s) | |
| 3. | Subject | Keyword(s) | directed first passage percolation, growth model, and phase transition |
| 3. | Subject | Subject classification | 60K35 |
| 4. | Description | Abstract | 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 $\vec{T}(0,(r,\theta))$ 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 $\vec{T}(0,(r,\theta))/r$ 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 $\vec{T}(0, (r, \theta))$ 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 $\vec{p}_c$ |
| 5. | Publisher | Organizing agency, location | |
| 6. | Contributor | Sponsor(s) | NSF grant DMS-4540247 |
| 7. | Date | (YYYY-MM-DD) | 2009-09-30 |
| 8. | Type | Status & genre | Peer-reviewed Article |
| 8. | Type | Type | |
| 9. | Format | File format | |
| 10. | Identifier | Uniform Resource Identifier | http://ejp.ejpecp.org/article/view/706 |
| 10. | Identifier | Digital Object Identifier | 10.1214/EJP.v14-706 |
| 11. | Source | Journal/conference title; vol., no. (year) | Electronic Journal of Probability; Vol 14 |
| 12. | Language | English=en | |
| 14. | Coverage | Geo-spatial location, chronological period, research sample (gender, age, etc.) | |
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