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April 2002 Limit theorems for the nonattractive Domany-Kinzel model
Makoto Katori, Norio Konno, Hideki Tanemura
Ann. Probab. 30(2): 933-947 (April 2002). DOI: 10.1214/aop/1023481012


We study the Domany–Kinzel model, which is a class of discrete time Markov processes with two parameters $(p_1, p_2) \in [0,1]^2$ and whose states are subsets of $\mathbf{Z}$, the set of integers. When $p_1 = \alpha \beta$ and $p_2 = \alpha (2 \beta - \beta^2)$ with $(\alpha, \beta) \in [0,1]^2$, the process can be identified with the mixed site–bond oriented percolation model on a square lattice with the probabilities of open site a and of open bond $\beta$. For the attractive case, $0 \leq p_1 \leq p_2 \leq 1$, the complete convergence theorem is easily obtained. On the other hand, the case $(p_1, p_2) = (1,0)$ realizes the rule 90 cellular automaton of Wolframin which, starting from the Bernoulli measure with density $\theta$, the distribution converges weakly only if $\theta \in {0, 1/2, 1}$. Using our new construction of processes based on signed measures, we prove limit theorems which are also valid for nonattractive cases with $(p_1, p_2) \not= (1,0)$. In particular, when $p_2 \in [0,1]$ and $p_1$ is close to 1, the complete convergence theorem is obtained as a corollary of the limit theorems.


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Makoto Katori. Norio Konno. Hideki Tanemura. "Limit theorems for the nonattractive Domany-Kinzel model." Ann. Probab. 30 (2) 933 - 947, April 2002.


Published: April 2002
First available in Project Euclid: 7 June 2002

zbMATH: 1018.60095
MathSciNet: MR1905861
Digital Object Identifier: 10.1214/aop/1023481012

Primary: 60K35 , 82B43 , 82C22

Keywords: Complete convergence theorem , Domany-Kinzel model , limit theorem , nonattractive process

Rights: Copyright © 2002 Institute of Mathematical Statistics


Vol.30 • No. 2 • April 2002
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