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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 thateach edge is open with probability $p f(y-x)$ where $f(y-x)$ is a fixednon-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 openoriented-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 shouldextend to the nearest neighbor-model for high-dimension, and for the spread-outmodel when $d=3,4$. It is known that this phenomenon does not occur indimension 1 and 2.

Existence of an intermediate phase for oriented percolation