The time constant vanishes only on the percolation cone in directed first passage percolation
| dc.creator | Zhang, Yu | |
| dc.date | 2008-02-24 | |
| dc.date | 2008-03-10 | |
| dc.date.accessioned | 2026-07-07T09:25:32Z | |
| dc.date.available | 2026-07-07T09:25:32Z | |
| dc.description | 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 $\vec{p}_c$. | |
| dc.description | 28 pages, 1 figure | |
| dc.identifier | https://arxiv.org/abs/0802.3519 | |
| dc.identifier | http://arxiv.org/abs/0802.3519 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/156435 | |
| dc.subject | Probability | |
| dc.subject | 60K35 | |
| dc.title | The time constant vanishes only on the percolation cone in directed first passage percolation | |
| dc.type | text |