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


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.


Download Citation

Itai Benjamini. Oded Schramm. "Recurrence of Distributional Limits of Finite Planar Graphs." Electron. J. Probab. 6 1 - 13, 2001.


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

Primary: 82B41
Secondary: 05C10 , 60J45

Keywords: Circle packing , mass trasport , random triangulations , Random walks , volume growth

Vol.6 • 2001
Back to Top