Asymptotic behavior for a long-range Domany-Kinzel model

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 p(k) is an element of[0, 1), where m is a non-negative integer. Let tau(m)(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 alpha = M/N, we derive the correct critical value alpha(m,c) is an element of R such that as N -> infinity, tau(m)(M, N) converges to 1, 0 and 1/2 for alpha > alpha(m.c), alpha < alpha(m.c) and alpha = alpha(m.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 p(n) is an element of[0, 1) with n is an element of Z(+) and for p(n)approximate to(n ->infinity)pn(-s) for p is an element of(0, 1) and s > 0. We find that the behavior of lim(m ->infinity) tau(m)(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). (C) 2018 Elsevier B.V. All rights reserved.