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2012 Existence of an intermediate phase for oriented percolation
Hubert Lacoin
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Electron. J. Probab. 17: 1-17 (2012). DOI: 10.1214/EJP.v17-1761

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.

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Hubert Lacoin. "Existence of an intermediate phase for oriented percolation." Electron. J. Probab. 17 1 - 17, 2012. https://doi.org/10.1214/EJP.v17-1761

Information

Accepted: 31 May 2012; Published: 2012
First available in Project Euclid: 4 June 2016

zbMATH: 1243.82037
MathSciNet: MR2928724
Digital Object Identifier: 10.1214/EJP.v17-1761

Subjects:
Primary: 82D60
Secondary: 60K37, 82B44

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