TY - JOUR

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

AU - Chang, Shu Chiuan

AU - Chen, Lung Chi

N1 - Funding Information:
We would like to thank anonymous referees for their useful suggestions. The research of S.C.C. was partially supported by NCTS at Taipei and the Ministry of Science and Technology grants MOST 102-2119-M-002-001 , MOST 100-2112-M-006-003-MY3 , and MOST 103-2918-I-006-016 . The research of L.C.C was partially supported by the mathematics division of the National Center for Theoretical Sciences (NCTS) and the Ministry of Science and Technology grant MOST 102-2115-M-004-005-MY2 .
Publisher Copyright:
© 2015 Elsevier B.V. All rights reserved.

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

AN - SCOPUS:84930948682

SN - 0378-4371

VL - 436

SP - 547

EP - 557

JO - Physica A: Statistical Mechanics and its Applications

JF - Physica A: Statistical Mechanics and its Applications

ER -