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Recurrence of Distributional Limits of Finite Planar Graphs


 
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1. Title Title of document Recurrence of Distributional Limits of Finite Planar Graphs
 
2. Creator Author's name, affiliation, country Itai Benjamini; The Weizmann Institute of Science
 
2. Creator Author's name, affiliation, country Oded Schramm; Microsoft Research
 
3. Subject Discipline(s) Mathematics
 
3. Subject Keyword(s) Random triangulations, random walks, mass trasport, circle packing, volume growth.
 
3. Subject Subject classification 82B41, 60J45 ,05C10.
 
4. Description 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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7. Date (YYYY-MM-DD) 2001-09-19
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ejp.ejpecp.org/article/view/96
 
10. Identifier Digital Object Identifier 10.1214/EJP.v6-96
 
11. Source Journal/conference title; vol., no. (year) Electronic Journal of Probability; Vol 6
 
12. Language English=en en
 
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