Limit theorems for the nonattractive Domany-Kinzel model

Abstract

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.