Recurrence of distributional limits of finite planar graphs. (English) Zbl 1010.82021

Benjamini, Itai; Schramm, Oded

Electron. J. Probab. 6, Paper No. 23, 13 p. (2001).

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

Keywords:

random triangulations; random walks; mass transport; circle packing; volume growth