Percolation on Triangulations: a Bijective Path to Liouville Quantum Gravity - Couverture souple

Bernardi, Olivier; Holden, Nina; Sun, Xin

Couverture souple

ISBN 10 : 1470466996 ISBN 13 : 9781470466992

Editeur : American Mathematical Society, 2023

Synopsis

We set the foundation for a series of works aimed at proving strong relations between uniform random planar maps and Liouville quantum gravity (LQG). Our method relies on a bijective encoding of site-percolated planar triangulations by certain 2D lattice paths. Our bijection parallels in the discrete setting the matingof-trees framework of LQG and Schramm-Loewner evolutions (SLE) introduced by Duplantier, Miller, and Sheffield. Combining these two correspondences allows us to relate uniform site-percolated triangulations to √8/3-LQG and SLE6. In particular, we establish the convergence of several functionals of the percolation model to continuous random objects defined in terms of √8/3-LQG and SLE6. For instance, we show that the exploration tree of the percolation converges to a branching SLE₆, and that the collection of percolation cycles converges to the conformal loop ensemble CLE₆. We also prove convergence of counting measure on the pivotal points of the percolation. Our results play an essential role in several other works, including a program for showing convergence of the conformal structure of uniform triangulations and works which study the behavior of random walk on the uniform infinite planar triangulation.

A propos du livre

À propos de l'auteur

Olivier Bernardi, Brandeis University, Waltham, Massachusetts.

Nina Holden, New York University, New York.

Xin Sun, Peking University, Beijing, China.

Détails de l'édition

• Éditeur: American Mathematical Society
• Date d'édition: 2023
• Langue: anglais
• ISBN 10: 1470466996
• ISBN 13: 9781470466992
• Reliure: Broché
• Nombre de pages: 176
• Coordonnées du fabricant: non disponible
• Personne responsable: non disponible