@article{EJP1761,
author = {Hubert Lacoin},
title = {Existence of an intermediate phase for oriented percolation},
journal = {Electron. J. Probab.},
fjournal = {Electronic Journal of Probability},
volume = {17},
year = {2012},
keywords = {Percolation: Growth model; Directed Polymers; Phase transition; Random media},
abstract = {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.},
pages = {no. 41, 1-17},
issn = {1083-6489},
doi = {10.1214/EJP.v17-1761},
url = {http://ejp.ejpecp.org/article/view/1761}}