Cover times for sequences of reversible Markov chains on random graphs

Abstract

We provide conditions that classify sequences of random graphs into two types in terms of cover times. One type (type $1$) is the class of random graphs on which the cover times are of the order of the maximal hitting times scaled by the logarithm of the size of vertex sets. The other type (type $2$) is the class of random graphs on which the cover times are of the order of the maximal hitting times. The conditions are described by some parameters determined by random graphs: the volumes, the diameters with respect to the resistance metric, and the coverings or packings by balls in the resistance metric. We apply the conditions to and classify a number of examples, such as supercritical Galton–Watson trees, the incipient infinite cluster of a critical Galton–Watson tree, and the Sierpinski gasket graph.