## 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 p_{k}∈[0,1), where m is a non-negative integer. Let τ_{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 α=M∕N, we derive the correct critical value α_{m,c}∈R such that as N→∞ τ_{m}(M,N) converges to 1, 0 and 1∕2 for α>α_{m,c}, α<α_{m,c} and α=α_{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}∈[0,1) with n∈Z_{+} and p_{n}≈_{n→∞}pn^{−s} for p∈(0,1) and s>0. We find that the behavior of lim_{m→∞}τ_{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).

Original language | English |
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Pages (from-to) | 112-127 |

Number of pages | 16 |

Journal | Physica A: Statistical Mechanics and its Applications |

Volume | 506 |

DOIs | |

Publication status | Published - 2018 Sep 15 |

## All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Condensed Matter Physics