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

Shu-Chiuan Chang, Lung Chi Chen

Research output: Contribution to journalArticle

3 Citations (Scopus)

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 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 ↑ ∞.

Original languageEnglish
Pages (from-to)500-522
Number of pages23
JournalJournal of Statistical Physics
Volume155
Issue number3
DOIs
Publication statusPublished - 2014 Jan 1

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Directed Percolation
Triangular Lattice
Asymptotic Behavior
Aspect Ratio
Alternate
Rate of Convergence
Horizontal
Vertical
Path
aspect ratio

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this

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title = "Asymptotic Behavior for a Version of Directed Percolation on the Triangular Lattice",
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 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 ↑ ∞.",
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Asymptotic Behavior for a Version of Directed Percolation on the Triangular Lattice. / Chang, Shu-Chiuan; Chen, Lung Chi.

In: Journal of Statistical Physics, Vol. 155, No. 3, 01.01.2014, p. 500-522.

Research output: Contribution to journalArticle

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