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

Shu-Chiuan Chang, Lung Chi Chen

Research output: Contribution to journalArticle

2 Citations (Scopus)

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)/(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 ↑ ∞.

Original languageEnglish
Pages (from-to)547-557
Number of pages11
JournalPhysica A: Statistical Mechanics and its Applications
Volume436
DOIs
Publication statusPublished - 2015 Jun 4

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Directed Percolation
Honeycomb
Asymptotic Behavior
bricks
Aspect Ratio
Alternate
Critical value
aspect ratio
Rate of Convergence
Horizontal
Vertical
Path

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Condensed Matter Physics

Cite this

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

In: Physica A: Statistical Mechanics and its Applications, Vol. 436, 04.06.2015, p. 547-557.

Research output: Contribution to journalArticle

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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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