The Annals of Probability

Scaling limits of random planar maps with a unique large face

Svante Janson and Sigurdur Örn Stefánsson

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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.

Article information

Source
Ann. Probab., Volume 43, Number 3 (2015), 1045-1081.

Dates
First available in Project Euclid: 5 May 2015

Permanent link to this document
https://projecteuclid.org/euclid.aop/1430830277

Digital Object Identifier
doi:10.1214/13-AOP871

Mathematical Reviews number (MathSciNet)
MR3342658

Zentralblatt MATH identifier
1320.05112

Subjects
Primary: 05C80: Random graphs [See also 60B20]
Secondary: 05C05: Trees 60F17: Functional limit theorems; invariance principles 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Keywords
Random maps planar maps mobiles simply generated trees continuum random tree Brownian tree

Citation

Janson, Svante; Stefánsson, Sigurdur Örn. Scaling limits of random planar maps with a unique large face. Ann. Probab. 43 (2015), no. 3, 1045--1081. doi:10.1214/13-AOP871. https://projecteuclid.org/euclid.aop/1430830277


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