Circular automata synchronize with high probability

Abstract In this paper we prove that a uniformly distributed random circular automaton A n of order n synchronizes with high probability (w.h.p.). More precisely, we prove that P [ A n  synchronizes ] = 1 − O ( 1 n ) . The main idea of the proof is to translate the synchronization problem into a problem concerning properties of a random matrix; these properties are then established with high probability by a careful analysis of the stochastic dependence structure among the random entries of the matrix. Additionally, we provide an upper bound for the probability of synchronization of circular automata in terms of chromatic polynomials of circulant graphs.