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 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).