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Title: Asymptotic Behavior for a Version of Directed Percolation on the Square Lattice
Authors: 陳隆奇
Chen, Lung-Chi
Contributors: 應數系
Keywords: Domany–Kinzel model;Directed percolation;Compact directed percolation;Asymptotic behavior;Critical behavior;Susceptibility;Exact solutions;Berry–Esseen theorem
Date: 2011.03
Issue Date: 2014-11-13 17:23:25 (UTC+8)
Abstract: We consider a version of directed bond percolation on a square lattice whose vertical edges are directed upward with probabilities pvpv and horizontal edges are directed rightward with probabilities phph and 1 in alternate rows. Let τ(M,N)τ(M,N) be the probability that there is a connected directed path of occupied edges from (0,0)(0,0) to (M,N)(M,N). For each View the MathML sourceph∈[0,1],pv=(0,1) and aspect ratio α=M/Nα=M/N fixed, it was established (Chen and Wu, 2006) [9] that there is an View the MathML sourceαc=[1−pv2−ph(1−pv)2]/2pv2 such that, as N→∞N→∞, τ(M,N)τ(M,N) is 11, 00, and 1/21/2 for α>αcα>αc, α<αcα<αc, and α=αcα=αc, respectively. In particular, for ph=0ph=0 or 11, 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)τ(M,N) and the asymptotic behavior of View the MathML sourceτ(Mn−,N) and View the MathML sourceτ(Mn+,N), where View the MathML sourceMn−/N↑αc and View the MathML sourceMn+/N↓αc as N↑∞N↑∞. Moreover, we obtain a susceptibility on the rectangular net {(m,n)∈Z+×Z+:0≤m≤M and 0≤n≤N}{(m,n)∈Z+×Z+:0≤m≤M and 0≤n≤N}. The proof is based on the Berry–Esseen theorem.
Relation: Physica A, 390(3), 419-426
Data Type: article
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