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 ) ∈ [0, 1] 2 and whose states are subsets of Z, the set of integers. 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 the probabilities of open site a and of open bond β. For the attractive case, 0 < p 1 ≥ p 2 < 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 Wolfram in which, starting from the Bernoulli measure with density 0, the distribution converges weakly only if θ E {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 ) ¬= (1, 0). In particular, when p 2 ∈ [0, 1] and p 1 is close to 1, the complete convergence theorem is obtained as a corollary of the limit theorems.