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 (p1, p2) ∈ [0, 1]2 and whose states are subsets of Z, the set of integers. 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 the probabilities of open site a and of open bond β. For the attractive case, 0 ≤ p1 ≤ p2 ≤ 1, the complete convergence theorem is easily obtained. On the other hand, the case (p1, p2) = (1,0) realizes the rule 90 cellular automaton of Wolfram in which, starting from the Bernoulli measure with density θ, the distribution converges weakly only if θ ∈ {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 (p1, p2) ∈ (1, 0). In particular, when p2 ∈ [0, 1] and p1 is close to 1, the complete convergence theorem is obtained as a corollary of the limit theorems.