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Existence of an intermediate phase for oriented percolation

Lacoin, Hubert (2012), Existence of an intermediate phase for oriented percolation, Electronic Journal of Probability, 17, p. 41. http://dx.doi.org/10.1214/EJP.v17-1761

Type
Article accepté pour publication ou publié
External document link
http://dx.doi.org/10.1214/EJP.v17-1761
Date
2012
Journal name
Electronic Journal of Probability
Volume
17
Publisher
Institute of Mathematical Statistics
Pages
41
Publication identifier
http://dx.doi.org/10.1214/EJP.v17-1761
Metadata
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Author(s)
Lacoin, Hubert
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.
Subjects / Keywords
percolation; growth model; directed polymers; phase transition; random media

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