Existence of an intermediate phase for oriented percolation

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 $pf(y-x)$ 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.

16 pages, 2 figures, further typos corrected, enlarged intro and bibliography