Recurrence of Distributional Limits of Finite Planar Graphs
Itai Benjamini, The Weizmann Institute of Science
Oded Schramm, Microsoft Research
Abstract
Suppose that Gj
is a sequence of finite connected planar graphs,
and in each Gj 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 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.
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