The Annals of Probability

Classification of scaling limits of uniform quadrangulations with a boundary

Erich Baur, Grégory Miermont, and Gourab Ray

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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.

Article information

Ann. Probab., Volume 47, Number 6 (2019), 3397-3477.

Received: August 2016
Revised: August 2018
First available in Project Euclid: 2 December 2019

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Mathematical Reviews number (MathSciNet)

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60F17: Functional limit theorems; invariance principles
Secondary: 05C80: Random graphs [See also 60B20]

Planar map quadrangulation Brownian map Brownian disk Brownian tree scaling limit Gromov–Hausdorff convergence


Baur, Erich; Miermont, Grégory; Ray, Gourab. Classification of scaling limits of uniform quadrangulations with a boundary. Ann. Probab. 47 (2019), no. 6, 3397--3477. doi:10.1214/18-AOP1316.

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Supplemental materials

  • Supplement to “Classification of scaling limits of uniform quadrangulations with a boundary.”. We provide the proofs of Lemmas 5.1, 5.2, 5.3 and 5.5, as well as the proof of Theorem 3.5, where the $\textsf{SCRT}$ appears in the limit.