We study the Domany-Kinzel model, which is a class of discrete-time Markov processes in one-dimension with two parameters (P1, P2) ∈ [0, 1]2. When P1 = αβ and P2 = α(2β-β2) with (α, β) ∈ [0, 1] 2, the process can be identified with the mixed site-bond oriented percolation model on a square lattice with probabilities α of a site being open and β of a bond being open. This paper treats dualities for the Domany-Kinzel model ξtA and the DKdual ηtA starting from A. We prove that (i) E(x |ξtA ∩ B|) = E(x|ξtB ∩ A|) if x = 1-(2P 1-P2)/P12, (ii) E(x |ξtA ∩ B|) = E(x|ηtB ∩ A|) if x = 1-(2P1-P2)/P1, and (iii) E(x |ηtA ∩ B|) = E(x|ηtB ∩ A|) if x = 1-(2P1-P2), as long as one of A, B is finite and P 2 ≤ P1.
|ジャーナル||Journal of Theoretical Probability|
|出版ステータス||Published - 2004 1|
ASJC Scopus subject areas
- 数学 (全般)