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2001 Recurrence of Distributional Limits of Finite Planar Graphs
Itai Benjamini, Oded Schramm
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Electron. J. Probab. 6: 1-13 (2001). DOI: 10.1214/EJP.v6-96

Abstract

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.

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Itai Benjamini. Oded Schramm. "Recurrence of Distributional Limits of Finite Planar Graphs." Electron. J. Probab. 6 1 - 13, 2001. https://doi.org/10.1214/EJP.v6-96

Information

Accepted: 19 September 2001; Published: 2001
First available in Project Euclid: 19 April 2016

zbMATH: 1010.82021
MathSciNet: MR1873300
Digital Object Identifier: 10.1214/EJP.v6-96

Subjects:
Primary: 82B41
Secondary: 05C10, 60J45

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Vol.6 • 2001
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