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 ↑ ∞.
UR - http://www.scopus.com/inward/record.url?scp=84930948682&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84930948682&partnerID=8YFLogxK
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 -