| Título: | Existence of an intermediate phase for oriented percolation |
| Autores: | Lacoin, Hubert; CNRS & Université Paris Dauphine |
| Fecha: | 2012-01-01 |
| Publicador: | Electronic journal of probability |
| Fuente: |
 |
| Tipo: | Peer-reviewed Article |
| Tema: |
Percolation: Growth model; Directed Polymers; Phase transition; Random media 82D60; 60K37; 82B44 |
| Descripción: | We consider the following oriented percolation model of $\mathbb {N} \times \mathbb{Z}^d$: we equip $\mathbb {N}\times \mathbb{Z}^d$ with the edge set $\{[(n,x),(n+1,y)] | n\in \mathbb {N}, x,y\in \mathbb{Z}^d\}$, and we say that each edge is open with probability $p f(y-x)$ where $f(y-x)$ is a fixed non-negative compactly supported function on $\mathbb{Z}^d$ with $\sum_{z\in \mathbb{Z}^d} f(z)=1$ and $p\in [0,\inf f^{-1}]$ is the percolation parameter. Let $p_c$ denote the percolation threshold ans $Z_N$ the number of open oriented-paths of length $N$ starting from the origin, and study the growth of $Z_N$ when percolation occurs. We prove that for if $d\ge 5$ and the function $f$ is sufficiently spread-out, then there exists a second threshold $p_c^{(2)}>p_c$ such that $Z_N/p^N$ decays exponentially fast for $p\in(p_c,p_c^{(2)})$ and does not so when $p> p_c^{(2)}$. The result should extend to the nearest neighbor-model for high-dimension, and for the spread-out model when $d=3,4$. It is known that this phenomenon does not occur in dimension 1 and 2. |
| Idioma: | Inglés |
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