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May 2015 Scaling limits of random planar maps with a unique large face
Svante Janson, Sigurdur Örn Stefánsson
Ann. Probab. 43(3): 1045-1081 (May 2015). DOI: 10.1214/13-AOP871

Abstract

We study random bipartite planar maps defined by assigning nonnegative weights to each face of a map. We prove that for certain choices of weights a unique large face, having degree proportional to the total number of edges in the maps, appears when the maps are large. It is furthermore shown that as the number of edges $n$ of the planar maps goes to infinity, the profile of distances to a marked vertex rescaled by $n^{-1/2}$ is described by a Brownian excursion. The planar maps, with the graph metric rescaled by $n^{-1/2}$, are then shown to converge in distribution toward Aldous’ Brownian tree in the Gromov–Hausdorff topology. In the proofs, we rely on the Bouttier–di Francesco–Guitter bijection between maps and labeled trees and recent results on simply generated trees where a unique vertex of a high degree appears when the trees are large.

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Svante Janson. Sigurdur Örn Stefánsson. "Scaling limits of random planar maps with a unique large face." Ann. Probab. 43 (3) 1045 - 1081, May 2015. https://doi.org/10.1214/13-AOP871

Information

Published: May 2015
First available in Project Euclid: 5 May 2015

zbMATH: 1320.05112
MathSciNet: MR3342658
Digital Object Identifier: 10.1214/13-AOP871

Subjects:
Primary: 05C80
Secondary: 05C05 , 60F17 , 60J80

Keywords: Brownian tree , Continuum random tree , mobiles , planar maps , Random maps , Simply generated trees

Rights: Copyright © 2015 Institute of Mathematical Statistics

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Vol.43 • No. 3 • May 2015
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