Open Access
November 2019 Classification of scaling limits of uniform quadrangulations with a boundary
Erich Baur, Grégory Miermont, Gourab Ray
Ann. Probab. 47(6): 3397-3477 (November 2019). DOI: 10.1214/18-AOP1316


We study noncompact scaling limits of uniform random planar quadrangulations with a boundary when their size tends to infinity. Depending on the asymptotic behavior of the boundary size and the choice of the scaling factor, we observe different limiting metric spaces. Among well-known objects like the Brownian plane or the self-similar continuum random tree, we construct two new one-parameter families of metric spaces that appear as scaling limits: the Brownian half-plane with skewness parameter $\theta$ and the infinite-volume Brownian disk of perimeter $\sigma$. We also obtain various coupling and limit results clarifying the relation between these objects.


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Erich Baur. Grégory Miermont. Gourab Ray. "Classification of scaling limits of uniform quadrangulations with a boundary." Ann. Probab. 47 (6) 3397 - 3477, November 2019.


Received: 1 August 2016; Revised: 1 August 2018; Published: November 2019
First available in Project Euclid: 2 December 2019

zbMATH: 07212165
MathSciNet: MR4038036
Digital Object Identifier: 10.1214/18-AOP1316

Primary: 60D05 , 60F17
Secondary: 05C80

Keywords: Brownian disk , Brownian map , Brownian tree , Gromov–Hausdorff convergence , Planar map , quadrangulation , Scaling limit

Rights: Copyright © 2019 Institute of Mathematical Statistics

Vol.47 • No. 6 • November 2019
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