Abstract (EN)

We consider the following oriented percolation model of N×Zd: we equip N×Zd with the edge set {[(n,x),(n+1,y)]|n∈N,x,y∈Zd}, 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 Zd with ∑z∈Zdf(z)=1 and p∈[0,inff−1] is the percolation parameter. Let pc denote the percolation threshold ans ZN the number of open oriented-paths of length N starting from the origin, and study the growth of ZN when percolation occurs. We prove that for if d≥5 and the function f is sufficiently spread-out, then there exists a second threshold p(2)c>pc such that ZN/pN decays exponentially fast for p∈(pc,p(2)c) and does not so when p>p(2)c. 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.