Limit theorems for the nonattractive Domany-Kinzel model

• 成果类型: Article
• 署名作者: Katori, M; Konno, N; Tanemura, H
• 署名单位: Chuo University; Yokohama National University; Chiba University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 2002
• 页码: 933-947
• 关键词: valued markov-processes; cellular-automata; percolation; survival

摘要

We study the Domany-Kinzel model, which is a class of discrete time Markov processes with two parameters (p(1), p(2)) is an element of [0, 1](2) and whose states are subsets of Z, the set of integers. When p(1) = alphabeta and p(2) = alpha(2beta - beta(2)) with (alpha, beta) is an element of [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 alpha and of open bond beta. For the attractive case, 0 less than or equal to p(1) less than or equal to p(2) less than or equal to 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 theta is an element of {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 equal (1, 0). In particular, when p(2) is an element of [0, 1] and p(1) is close to 1, the complete convergence theorem is obtained as a corollary of the limit theorems.