Dualities for the Domany-Kinzel model

Makoto Katori, Norio Konno, Aidan Sudbury, Hideki Tanemura

Research output: Contribution to journalArticle

4 Citations (Scopus)

Abstract

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.

Original languageEnglish
Pages (from-to)131-144
Number of pages14
JournalJournal of Theoretical Probability
Volume17
Issue number1
DOIs
Publication statusPublished - 2004 Jan 1
Externally publishedYes

Fingerprint

Duality
Oriented Percolation
Square Lattice
Markov Process
One Dimension
Two Parameters
Discrete-time
Model
Class
Markov process

Keywords

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

ASJC Scopus subject areas

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

Cite this

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

In: Journal of Theoretical Probability, Vol. 17, No. 1, 01.01.2004, p. 131-144.

Research output: Contribution to journalArticle

Katori, Makoto ; Konno, Norio ; Sudbury, Aidan ; Tanemura, Hideki. / Dualities for the Domany-Kinzel model. In: Journal of Theoretical Probability. 2004 ; Vol. 17, No. 1. pp. 131-144.
@article{3f5c8f9b8a0545edba06651270eee67b,
title = "Dualities for the Domany-Kinzel model",
abstract = "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.",
keywords = "Duality, The DK dual, The Domany-Kinzel model",
author = "Makoto Katori and Norio Konno and Aidan Sudbury and Hideki Tanemura",
year = "2004",
month = "1",
day = "1",
doi = "10.1023/B:JOTP.0000020478.24536.26",
language = "English",
volume = "17",
pages = "131--144",
journal = "Journal of Theoretical Probability",
issn = "0894-9840",
publisher = "Springer New York",
number = "1",

}

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 -