Random majority percolation

We shall consider the discrete time synchronous random majority-vote cellular automata on the n by n torus, in which every vertex is in one of two states and, at each time step t, every vertex goes into the state the majority of its neighbours had at time t−1 with a small chance p of error independently of all other events. We shall show that, if n is fixed and p is sufficiently small, then the process spends almost half of its time in each of two configurations. Further, we show that the expected time for it to reach one of these configurations from the other is Θ(1/pn+1) despite the actual time spent in transit being O(1/p3). Unusually, it appears difficult to obtain any results for this regime by simulation. In this paper we shall consider various cellular automata on graphs. Cellular automata with random initial configuration and deterministic update rules have been studied in numerous papers (see the following section for a discussion of previous work), however, our emphasis in this paper is on automata with probabilistic update rules, i.e., rules that are applied with probability less than one. Such cellular automata are often refered to as ∗Research supported by NSF grant EIA-0130352. †University of Memphis, Department of Mathematics, Dunn Hall, 3725 Noriswood, Memphis, TN 38152, USA. ‡Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge CB3 0WB, UK §Research supported in part by NSF grants DMS-0505550, CNS-0721983 and CCF-0728928, and ARO grant W911NF-06-1-0076 ¶Queen Mary University of London, London E1 4NS. UK