### 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 |

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### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Condensed Matter Physics

### Cite this

*Physica A: Statistical Mechanics and its Applications*,

*506*, 112-127. https://doi.org/10.1016/j.physa.2018.03.061

}

*Physica A: Statistical Mechanics and its Applications*, vol. 506, pp. 112-127. https://doi.org/10.1016/j.physa.2018.03.061

**Asymptotic behavior for a long-range Domany–Kinzel model.** / Chang, Shu-Chiuan; Chen, Lung Chi.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Asymptotic behavior for a long-range Domany–Kinzel model

AU - Chang, Shu-Chiuan

AU - Chen, Lung Chi

PY - 2018/9/15

Y1 - 2018/9/15

N2 - 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).

AB - 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).

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UR - http://www.scopus.com/inward/citedby.url?scp=85045664691&partnerID=8YFLogxK

U2 - 10.1016/j.physa.2018.03.061

DO - 10.1016/j.physa.2018.03.061

M3 - Article

VL - 506

SP - 112

EP - 127

JO - Physica A: Statistical Mechanics and its Applications

JF - Physica A: Statistical Mechanics and its Applications

SN - 0378-4371

ER -