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February 2021 Liouville quantum gravity surfaces with boundary as matings of trees
Morris Ang, Ewain Gwynne
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Ann. Inst. H. Poincaré Probab. Statist. 57(1): 1-53 (February 2021). DOI: 10.1214/20-AIHP1068


For γ(0,2), the quantum disk and γ-quantum wedge are two of the most natural types of Liouville quantum gravity (LQG) surfaces with boundary. These surfaces arise as scaling limits of finite and infinite random planar maps with boundary, respectively. We show that the left/right quantum boundary length process of a space-filling SLE16/γ2 curve on a quantum disk or on a γ-quantum wedge is a certain explicit conditioned two-dimensional Brownian motion with correlation cos(πγ2/4). This extends the mating of trees theorem of Duplantier, Miller, and Sheffield (2014) to the case of quantum surfaces with boundary (the disk case for γ(2,2) was previously treated by Duplantier, Miller, Sheffield using different methods). As an application, we give an explicit formula for the conditional law of the LQG area of a quantum disk given its boundary length by computing the law of the corresponding functional of the correlated Brownian motion.

Pour γ(0,2), le disque quantique et le γ-secteur angulaire quantique sont deux types, parmi les plus naturels, de surfaces avec frontières pour la gravité quantique de Liouville (LQG). Ces surfaces apparaissent comme limites d’échelle des cartes planaires, respectivement finies et infinies, avec frontières. Nous montons que les processus des longueurs de la frontière quantique à gauche/droite d’une courbe SLE16/γ2 sur un disque quantique ou un γ-secteur angulaire quantique est un mouvement Brownien 2-dimensionnel, sous un conditionnement explicite, avec corrélation cos(πγ2/4). Ceci étend le théorème d’accouplement d’arbres de Duplantier, Miller, et Sheffield (2014) au cas des surfaces quantiques avec frontières (le cas du disque pour γ(2,2) avait été traité par Duplantier, Miller, Sheffield en utilisant des méthodes différentes). Comme application, nous donnons une formule explicite pour la loi conditionnelle de l’aire de la LQG d’un disque quantique étant donnée la longueur de sa frontière en calculant la loi de la fonctionnelle correspondante du mouvement Brownien corrélé.


We thank Jason Miller, Minjae Park, Guillaume Remy, Scott Sheffield, and Xin Sun for helpful discussions. We thank an anonymous referee for helpful comments on an earlier version of this paper. We also thank the Isaac Newton Institute for Mathematical Sciences, Cambridge University, for its hospitality during the Random Geometry Workshop where part of this work was carried out. M.A. was partially supported by the NSF grant DMS-1712862. E.G. was partially supported by a Herchel Smith fellowship and a Trinity College junior research fellowship.


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Morris Ang. Ewain Gwynne. "Liouville quantum gravity surfaces with boundary as matings of trees." Ann. Inst. H. Poincaré Probab. Statist. 57 (1) 1 - 53, February 2021.


Received: 2 April 2019; Revised: 21 April 2020; Accepted: 7 May 2020; Published: February 2021
First available in Project Euclid: 12 March 2021

Digital Object Identifier: 10.1214/20-AIHP1068

Primary: 60D05, 60J65, 60J67

Rights: Copyright © 2021 Association des Publications de l’Institut Henri Poincaré


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Vol.57 • No. 1 • February 2021
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