Asymptotic behavior for a long-range Domany–Kinzel model

Shu-Chiuan Chang and Lung-Chi Chen

Physica A: Statistical Mechanics and its Applications, 2018, vol. 506, issue C, 112-127

Abstract: We consider a long-range Domany–Kinzel model proposed by Li and Zhang (1983), such that for every site (i,j) in a two-dimensional rectangular lattice there is a directed bond present from site (i,j) to (i+1,j) with probability one. There are also m+1 directed bounds present from (i,j) to (i−k+1,j+1), k=0,1,…,m with probability pk∈[0,1), where m is a non-negative integer. Let τm(M,N) be the probability that there is at least one connected-directed path of occupied edges from (0,0) to (M,N). Defining the aspect ratio α=M∕N, we derive the correct critical value αm,c∈R such that as N→∞, τm(M,N) converges to 1, 0 and 1∕2 for α>αm,c, α<αm,c and α=αm,c, respectively, and we study the rate of convergence. Furthermore, we investigate the cases in the infinite m limit. Specifically, we discuss in details the case such that pn∈[0,1) with n∈Z+ and pn≈n→∞pn−s for p∈(0,1) and s>0. We find that the behavior of limm→∞τm(M,N) for this case highly depends on the value of s and how fast one approaches to the critical aspect ratio. The present study corrects and extends the results given in Li and Zhang (1983).

Keywords: Domany–Kinzel model; Directed percolation; Random walk; Asymptotic behavior; Critical behavior; Berry–Esseen theorem; Large deviation (search for similar items in EconPapers)
Date: 2018
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Persistent link: https://EconPapers.repec.org/RePEc:eee:phsmap:v:506:y:2018:i:c:p:112-127

DOI: 10.1016/j.physa.2018.03.061

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