A nucleation–growth process on the integers

Abstract

We consider a Markov process with state space {0, 1}Z where Os become 1s irreversibly at rates which depend on whether none of a 0&s nearest neighbours (nucleation), its left-hand neighbour (right-hand growth), or its right-hand neighbour (left-hand growth) is in the 1-state. Furthermore, we assume that Os with both nearest neighbours in the 1-state remain in the 0-state forever and that at time 0 only the origin is in the 1-state. We determine the size distribution of the cluster (maximal sequence of 1s uninterrupted by Os) at the origin and the distribution of the time when its growth is stopped (by birth (nucleation) or competitive growth of neighbouring clusters). In the final state of the process, the spatial distribution of the (trapped) sites in the 0-state is considered. Some information on the size distribution of clusters well away from the cluster at the origin is obtained.