Asymptotic behavior for a version of directed percolation on a square lattice

Lung-Chi Chen

Physica A: Statistical Mechanics and its Applications, 2011, vol. 390, issue 3, 419-426

Abstract: We consider a version of directed bond percolation on a square lattice whose vertical edges are directed upward with probabilities pv and horizontal edges are directed rightward with probabilities ph and 1 in alternate rows. Let τ(M,N) be the probability that there is a connected directed path of occupied edges from (0,0) to (M,N). For each ph∈[0,1],pv=(0,1) and aspect ratio α=M/N fixed, it was established (Chen and Wu, 2006) [9] that there is an αc=[1−pv2−ph(1−pv)2]/2pv2 such that, as N→∞, τ(M,N) is 1, 0, and 1/2 for α>αc, α<αc, and α=αc, respectively. In particular, for ph=0 or 1, the model reduces to the Domany–Kinzel model (Domany and Kinzel, 1981 [7]). In this article, we investigate the rate of convergence of τ(M,N) and the asymptotic behavior of τ(Mn−,N) and τ(Mn+,N), where Mn−/N↑αc and Mn+/N↓αc as N↑∞. Moreover, we obtain a susceptibility on the rectangular net {(m,n)∈Z+×Z+:0≤m≤M and 0≤n≤N}. The proof is based on the Berry–Esseen theorem.

Keywords: Domany–Kinzel model; Directed percolation; Compact directed percolation; Asymptotic behavior; Critical behavior; Susceptibility; Exact solutions; Berry–Esseen theorem (search for similar items in EconPapers)
Date: 2011
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Persistent link: https://EconPapers.repec.org/RePEc:eee:phsmap:v:390:y:2011:i:3:p:419-426

DOI: 10.1016/j.physa.2010.09.039

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