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 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 |
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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 journal › Article
}
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 -