# zbMATH — the first resource for mathematics

Recurrence of distributional limits of finite planar graphs. (English) Zbl 1010.82021
Summary: Suppose that $$G_j$$ is a sequence of finite connected planar graphs, and in each $$G_j$$ a special vertex, called the root, is chosen randomly-uniformly. We introduce the notion of a distributional limit $$G$$ of such graphs. Assume that the vertex degrees of the vertices in $$G_j$$ 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.

##### MSC:
 82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics 60J45 Probabilistic potential theory 05C10 Planar graphs; geometric and topological aspects of graph theory
Full Text: