Random Walks , Liouville ' S Theorem , and Circle Packings

It has been shown that univalent circle packings lling in the complex plane C are unique up to similarities of C. Here we prove that bounded degree branched circle packings properly covering C are uniquely determined, up to similarities of C, by their branch sets. In particular, when branch sets of the packings considered are empty we obtain the earlier result. We also establish a circle packing analogue of Liouville's theorem: if f is a circle packing map whose domain packing is innnite, univalent, and has recurrent tangency graph, then the ratio map associated with f is either unbounded or constant.