Electronic Journal of Probability

Recurrence of Distributional Limits of Finite Planar Graphs

Itai Benjamini and Oded Schramm

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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.

Article information

Electron. J. Probab. Volume 6 (2001), paper no. 23, 13 pp.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 82B41: Random walks, random surfaces, lattice animals, etc. [See also 60G50, 82C41]
Secondary: 60J45 05C10: Planar graphs; geometric and topological aspects of graph theory [See also 57M15, 57M25]

Random triangulations random walks mass trasport circle packing volume growth

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Benjamini, Itai; Schramm, Oded. Recurrence of Distributional Limits of Finite Planar Graphs. Electron. J. Probab. 6 (2001), paper no. 23, 13 pp. doi:10.1214/EJP.v6-96. https://projecteuclid.org/euclid.ejp/1461097653

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