Persistence of sums of correlated increments and clustering in cellular automata

Abstract

Let T be the first return time to (−∞,0] of sums of increments given by a functional of a stationary Markov chain. We determine the asymptotic behavior of the survival probability, P(T≥t)∼Ct−1∕2 for an explicit constant C. Our analysis is based on a connection between the survival probability and the running maximum of the time-reversed process, and relies on a functional central limit theorem for Markov chains. As applications, we recover known clustering results for the 3-color cyclic cellular automaton and the Greenberg–Hastings model, and we prove a new clustering result for the 3-color firefly cellular automaton.