Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Scaling limits for the critical Fortuin–Kasteleyn model on a random planar map I: Cone times

Ewain Gwynne, Cheng Mao, and Xin Sun

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Abstract

Sheffield (2011) introduced an inventory accumulation model which encodes a random planar map decorated by a collection of loops sampled from the critical Fortuin–Kasteleyn (FK) model. He showed that a certain two-dimensional random walk associated with the infinite-volume version of the model converges in the scaling limit to a correlated planar Brownian motion. We improve on this scaling limit result by showing that the times corresponding to FK loops (or “flexible orders”) in the inventory accumulation model converge in the scaling limit to the $\pi/2$-cone times of the correlated Brownian motion. This statement implies a scaling limit result for the joint law of the areas and boundary lengths of the bounded complementary connected components of the FK loops on the infinite-volume planar map. In light of the encoding of Duplantier, Miller, and Sheffield (2014), the limiting object coincides with the joint law of the areas and boundary lengths of the bounded complementary connected components of a collection of CLE loops on an independent Liouville quantum gravity surface.

Résumé

Sheffield a introduit en 2011 un modèle d’accumulation de stocks, qui code une carte planaire aléatoire décorée par une collection de boucles, échantillonnée selon le modèle de percolation de Fortuin–Kasteleyn (FK) critique. Il a démontré que certaines marches aléatoires planes associées au modèle en volume infini convergent dans la limite d’échelle vers un mouvement brownien plan corrélé. Nous améliorons ce résultat de limite d’échelle en montrant que les temps correspondant aux boucles FK (ou « commandes flexibles ») dans le modèle d’accumulation de stocks convergent dans la limite d’échelle vers les temps de cône d’angle $\pi/2$ du mouvement brownien limite. Cet énoncé implique un résultat de limite d’échelle pour la loi jointe des aires et des longueurs de bord des composantes connexes bornées du complémentaire des boucles FK sur la carte de volume infini. À la lumière du codage de Duplantier, Miller et Sheffield (2014), l’objet limite coïncide avec la loi jointe des aires et des longueurs de bords des composantes connexes bornées du complémentaire d’une collection de boucles CLE sur une surface indépendante dont la loi est donnée par la gravité quantique de Liouville.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 55, Number 1 (2019), 1-60.

Dates
Received: 9 August 2016
Revised: 19 August 2017
Accepted: 12 November 2017
First available in Project Euclid: 18 January 2019

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1547802394

Digital Object Identifier
doi:10.1214/17-AIHP874

Subjects
Primary: 60F17: Functional limit theorems; invariance principles 60G50: Sums of independent random variables; random walks
Secondary: 82B27: Critical phenomena

Keywords
Fortuin–Kasteleyn model Random planar maps Hamburger–cheeseburger bijection Random walks in cones Liouville quantum gravity Schramm–Loewner evolution Conformal loop ensembles Peanosphere

Citation

Gwynne, Ewain; Mao, Cheng; Sun, Xin. Scaling limits for the critical Fortuin–Kasteleyn model on a random planar map I: Cone times. Ann. Inst. H. Poincaré Probab. Statist. 55 (2019), no. 1, 1--60. doi:10.1214/17-AIHP874. https://projecteuclid.org/euclid.aihp/1547802394


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