CELLULAR AUTOMATA MODELS OF RING DYNAMICS janko gravner

This paper describes three models arising from the theory of excitable media, whose primary visual feature are expanding rings of excitation. Rigorous mathematical results and experimental/computational issues are both addressed. We start with the much{studied Greenberg{ Hastings model (GHM) in which the rings are very short{lived, but they do have a transient percolation property. By contrast, in the model we call annihilating nested rings (ANR), excitation centers only gradually lose strength, i.e. each time they become inactive (and then stay so forever) with a xed probability; we show how the long{term global connectivity properties of the set of excited sites undergo a phase transition. Second part of the paper is devoted to digital boiling (DB) in which new rings spontaneously appear at rested sites with a positive probability. We focus on such (related) issues as convergence to equilibrium, equilibrium excitation level and success of the basic coupling.