### Abstract

We consider a version of directed bond percolation on the triangular lattice such that vertical edges are directed upward with probability y, diagonal edges are directed from lower-left to upper-right or lower-right to upper-left with probability d, and horizontal edges are directed rightward with probabilities x and one in alternate rows. Let t(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), d ∈ [0, 1) but (1 - y)(1 - d) ≠ 1 and aspect ratio α = M/N fixed for the triangular lattice with diagonal edges from lower-left to upper-right, we show that there is an α_{c} = (d-y-dy)/[2(d+y-dy)]+[1-(1-d)^{2}(1-y)^{2}x]/[2(d+ y - dy)^{2}] such that as N → ∞, τ(M, N) is 1, 0 and 1/2 for α > α_{c}, α < α_{c} and α = α_{c}, respectively. A corresponding result is obtained for the triangular lattice with diagonal edges from lower-right to upper-left. We also investigate the rate of convergence of τ(M, N) and the asymptotic behavior of t(M^{-}_{N}, N) and τ(M^{+}_{N}, N) where M^{-}_{N}/N ↑ α_{c} and M^{+}_{N}/N ↓ α_{c} as N ↑ ∞.

Original language | English |
---|---|

Pages (from-to) | 500-522 |

Number of pages | 23 |

Journal | Journal of Statistical Physics |

Volume | 155 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2014 Jan 1 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Journal of Statistical Physics*,

*155*(3), 500-522. https://doi.org/10.1007/s10955-014-0961-7

}

*Journal of Statistical Physics*, vol. 155, no. 3, pp. 500-522. https://doi.org/10.1007/s10955-014-0961-7

**Asymptotic Behavior for a Version of Directed Percolation on the Triangular 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 Triangular Lattice

AU - Chang, Shu-Chiuan

AU - Chen, Lung Chi

PY - 2014/1/1

Y1 - 2014/1/1

N2 - We consider a version of directed bond percolation on the triangular lattice such that vertical edges are directed upward with probability y, diagonal edges are directed from lower-left to upper-right or lower-right to upper-left with probability d, and horizontal edges are directed rightward with probabilities x and one in alternate rows. Let t(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), d ∈ [0, 1) but (1 - y)(1 - d) ≠ 1 and aspect ratio α = M/N fixed for the triangular lattice with diagonal edges from lower-left to upper-right, we show that there is an αc = (d-y-dy)/[2(d+y-dy)]+[1-(1-d)2(1-y)2x]/[2(d+ y - dy)2] such that as N → ∞, τ(M, N) is 1, 0 and 1/2 for α > αc, α < αc and α = αc, respectively. A corresponding result is obtained for the triangular lattice with diagonal edges from lower-right to upper-left. We also investigate the rate of convergence of τ(M, N) and the asymptotic behavior of t(M-N, N) and τ(M+N, N) where M-N/N ↑ αc and M+N/N ↓ αc as N ↑ ∞.

AB - We consider a version of directed bond percolation on the triangular lattice such that vertical edges are directed upward with probability y, diagonal edges are directed from lower-left to upper-right or lower-right to upper-left with probability d, and horizontal edges are directed rightward with probabilities x and one in alternate rows. Let t(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), d ∈ [0, 1) but (1 - y)(1 - d) ≠ 1 and aspect ratio α = M/N fixed for the triangular lattice with diagonal edges from lower-left to upper-right, we show that there is an αc = (d-y-dy)/[2(d+y-dy)]+[1-(1-d)2(1-y)2x]/[2(d+ y - dy)2] such that as N → ∞, τ(M, N) is 1, 0 and 1/2 for α > αc, α < αc and α = αc, respectively. A corresponding result is obtained for the triangular lattice with diagonal edges from lower-right to upper-left. We also investigate the rate of convergence of τ(M, N) and the asymptotic behavior of t(M-N, N) and τ(M+N, N) where M-N/N ↑ αc and M+N/N ↓ αc as N ↑ ∞.

UR - http://www.scopus.com/inward/record.url?scp=84898540955&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84898540955&partnerID=8YFLogxK

U2 - 10.1007/s10955-014-0961-7

DO - 10.1007/s10955-014-0961-7

M3 - Article

VL - 155

SP - 500

EP - 522

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 3

ER -