TY - JOUR

T1 - Asymptotic Behavior for a Version of Directed Percolation on the Triangular Lattice

AU - Chang, Shu Chiuan

AU - Chen, Lung Chi

N1 - Funding Information:
Acknowledgments We would like to thank Rongfeng Sun for many useful suggestions. The research of S.C.C. was partially supported by the National Science Council grants NSC-97-2112-M-006-007-MY3, NSC-100-2119-M-002-001 and NSC-100-2112-M-006-003-MY3. The research of L.C.C was partially supported by NCTS and the NSC grants NSC-99-2115-M-030-004-MY3 and NSC-102-2115-M-030-001-MY2.

PY - 2014/5

Y1 - 2014/5

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 ↑ ∞.

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U2 - 10.1007/s10955-014-0961-7

DO - 10.1007/s10955-014-0961-7

M3 - Article

AN - SCOPUS:84898540955

SN - 0022-4715

VL - 155

SP - 500

EP - 522

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

IS - 3

ER -