### Abstract

We consider a version of directed bond percolation on the honeycomb lattice as a brick lattice such that vertical edges are directed upward with probability y, and horizontal edges are directed rightward with probabilities x and one in alternate rows. Let τ(M,N) be the probability that there is at least one connected-directed path of occupied edges from (0,0) to (M,N). For each x∈(0,1], y∈(0,1] and aspect ratio α=M/N fixed, we show that there is a critical value ^{αc}=(1-x+xy)(1+x-xy)/(x^{y2}) such that as N→∞, τ(M,N) is 1, 0 and 1/2 for α>^{αc}, α<^{αc} and α=^{αc}, respectively. We also investigate the rate of convergence of τ(M,N) and the asymptotic behavior of τ(MN-,_{N}) and τ(M_{N}+,N) where M_{N}-/N ↑ α_{c} and M_{N}+/N↓^{αc} as N ↑ ∞.

Original language | English |
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Pages (from-to) | 547-557 |

Number of pages | 11 |

Journal | Physica A: Statistical Mechanics and its Applications |

Volume | 436 |

DOIs | |

Publication status | Published - 2015 Jun 4 |

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### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Condensed Matter Physics

### Cite this

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*Physica A: Statistical Mechanics and its Applications*, vol. 436, pp. 547-557. https://doi.org/10.1016/j.physa.2015.05.083

**Asymptotic behavior for a version of directed percolation on the honeycomb lattice.** / Chang, Shu-Chiuan; Chen, Lung Chi.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Asymptotic behavior for a version of directed percolation on the honeycomb lattice

AU - Chang, Shu-Chiuan

AU - Chen, Lung Chi

PY - 2015/6/4

Y1 - 2015/6/4

N2 - We consider a version of directed bond percolation on the honeycomb lattice as a brick lattice such that vertical edges are directed upward with probability y, and horizontal edges are directed rightward with probabilities x and one in alternate rows. Let τ(M,N) be the probability that there is at least one connected-directed path of occupied edges from (0,0) to (M,N). For each x∈(0,1], y∈(0,1] and aspect ratio α=M/N fixed, we show that there is a critical value αc=(1-x+xy)(1+x-xy)/(xy2) such that as N→∞, τ(M,N) is 1, 0 and 1/2 for α>αc, α<αc and α=αc, respectively. We also investigate the rate of convergence of τ(M,N) and the asymptotic behavior of τ(MN-,N) and τ(MN+,N) where MN-/N ↑ αc and MN+/N↓αc as N ↑ ∞.

AB - We consider a version of directed bond percolation on the honeycomb lattice as a brick lattice such that vertical edges are directed upward with probability y, and horizontal edges are directed rightward with probabilities x and one in alternate rows. Let τ(M,N) be the probability that there is at least one connected-directed path of occupied edges from (0,0) to (M,N). For each x∈(0,1], y∈(0,1] and aspect ratio α=M/N fixed, we show that there is a critical value αc=(1-x+xy)(1+x-xy)/(xy2) such that as N→∞, τ(M,N) is 1, 0 and 1/2 for α>αc, α<αc and α=αc, respectively. We also investigate the rate of convergence of τ(M,N) and the asymptotic behavior of τ(MN-,N) and τ(MN+,N) where MN-/N ↑ αc and MN+/N↓αc as N ↑ ∞.

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U2 - 10.1016/j.physa.2015.05.083

DO - 10.1016/j.physa.2015.05.083

M3 - Article

VL - 436

SP - 547

EP - 557

JO - Physica A: Statistical Mechanics and its Applications

JF - Physica A: Statistical Mechanics and its Applications

SN - 0378-4371

ER -