### 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 |
---|---|

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 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

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

### Cite this

*Journal of Theoretical Probability*,

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

**Dualities for the Domany-Kinzel model.** / Katori, Makoto; Konno, Norio; Sudbury, Aidan; Tanemura, Hideki.

Research output: Contribution to journal › Article

*Journal of Theoretical Probability*, vol. 17, no. 1, pp. 131-144. https://doi.org/10.1023/B:JOTP.0000020478.24536.26

}

TY - JOUR

T1 - Dualities for the Domany-Kinzel model

AU - Katori, Makoto

AU - Konno, Norio

AU - Sudbury, Aidan

AU - Tanemura, Hideki

PY - 2004/1/1

Y1 - 2004/1/1

N2 - 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.

AB - 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.

KW - Duality

KW - The DK dual

KW - The Domany-Kinzel model

UR - http://www.scopus.com/inward/record.url?scp=4043092815&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=4043092815&partnerID=8YFLogxK

U2 - 10.1023/B:JOTP.0000020478.24536.26

DO - 10.1023/B:JOTP.0000020478.24536.26

M3 - Article

AN - SCOPUS:4043092815

VL - 17

SP - 131

EP - 144

JO - Journal of Theoretical Probability

JF - Journal of Theoretical Probability

SN - 0894-9840

IS - 1

ER -