Recurrence of Distributional Limits of Finite Planar Graphs

Suppose that Gj is a sequence of finite connected planar graphs, and in each Gj a special vertex, called the root, is chosen randomlyuniformly. We introduce the notion of a distributional limit G of such graphs. Assume that the vertex degrees of the vertices in Gj are bounded, and the bound does not depend on j. Then after passing to a subsequence, the limit exists, and is a random rooted graph G. We prove that with probability one G is recurrent. The proof involves the Circle Packing Theorem. The motivation for this work comes from the theory of random spherical triangulations.