Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Liouville quantum gravity on the unit disk

Yichao Huang, Rémi Rhodes, and Vincent Vargas

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Our purpose is to pursue the rigorous construction of Liouville Quantum Field Theory on Riemann surfaces initiated by F. David, A. Kupiainen and the last two authors in the context of the Riemann sphere and inspired by the 1981 seminal work by Polyakov. In this paper, we investigate the case of simply connected domains with boundary. We also make precise conjectures about the relationship of this theory to scaling limits of random planar maps with boundary conformally embedded onto the disk.


Notre but est d’étendre la construction rigoureuse de la Théorie Quantique des Champs de Liouville sur les surfaces de Riemann, initiée par F. David, A. Kupiainen et les deux derniers auteurs dans le contexte de la sphère de Riemann et inspirée par le travail pionnier de Polyakov en 1981. Dans ce papier nous étudions la théorie dans le cas de domaines simplement connexes à bord. Nous formulons également des conjectures précises sur la relation entre cette théorie et les limites d’échelle des grandes cartes planaires aléatoires à bord conformément plongées dans le disque unité.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 54, Number 3 (2018), 1694-1730.

Received: 2 October 2015
Revised: 27 June 2017
Accepted: 9 July 2017
First available in Project Euclid: 11 July 2018

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Zentralblatt MATH identifier

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 81T40: Two-dimensional field theories, conformal field theories, etc. 81T20: Quantum field theory on curved space backgrounds

Liouville Quantum Gravity Quantum field theory Gaussian multiplicative chaos KPZ formula KPZ scaling laws Polyakov formula Conformal anomaly


Huang, Yichao; Rhodes, Rémi; Vargas, Vincent. Liouville quantum gravity on the unit disk. Ann. Inst. H. Poincaré Probab. Statist. 54 (2018), no. 3, 1694--1730. doi:10.1214/17-AIHP852.

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