2012/01/22 by Hubert Lacoin, Lacoin, Hubert
Mathematics · Physics and Astronomy · #60K37 #82B44 #82D60 #FOS: Mathematics #FOS: Physical sciences #Markov Chains and Monte Carlo Methods #Mathematical Physics (math-ph) #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #math-ph #math.MP #math.PR #msc:60K37 #msc:82B44 #msc:82D60
paper · pdf · doi:10.48550/arxiv.1201.4552
16 pages, 2 figures, further typos corrected, enlarged intro and bibliography
openalex publication_date 2012/01/22 · arxiv created 2012/02/07 · arxiv updated 2012/02/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the following oriented percolation model of \mathbb N × ℤd: we equip \mathbb N× ℤd with the edge set \[(n,x),(n+1,y)] | n∈ \mathbb N, x,y∈ ℤd\, 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 ℤd with ∑z∈ ℤd f(z)=1 and p∈ [0,inf f-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 pc(2)>pc such that ZN/pN decays exponentially fast for p∈(pc,pc(2)) and does not so when p> pc(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.