### Abstract

We study the Domany-Kinzel model, which is a class of discrete-time Markov processes in one-dimension with two parameters (P_{1}, P_{2}) ∈ [0, 1]^{2}. When P_{1} = αβ and P_{2} = α(2β-β^{2}) 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 ξ_{t}^{A} and the DKdual η_{t}^{A} starting from A. We prove that (i) E(x ^{|ξtA ∩ B|}) = E(x^{|ξtB ∩ A|}) if x = 1-(2P _{1}-P_{2})/P_{1}^{2}, (ii) E(x ^{|ξtA ∩ B|}) = E(x^{|ηtB ∩ A|}) if x = 1-(2P_{1}-P_{2})/P_{1}, and (iii) E(x ^{|ηtA ∩ B|}) = E(x^{|ηtB ∩ A|}) if x = 1-(2P_{1}-P_{2}), as long as one of A, B is finite and P _{2} ≤ P_{1}.

Original language | English |
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Pages (from-to) | 131-144 |

Number of pages | 14 |

Journal | Journal of Theoretical Probability |

Volume | 17 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2004 Jan 1 |

Externally published | Yes |

### Keywords

- Duality
- The DK dual
- The Domany-Kinzel model

### ASJC Scopus subject areas

- Statistics and Probability
- Mathematics(all)
- Statistics, Probability and Uncertainty

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## Cite this

*Journal of Theoretical Probability*,

*17*(1), 131-144. https://doi.org/10.1023/B:JOTP.0000020478.24536.26