The three-point function of planar quadrangulations

2008/05/31 by J. Bouttier, Jérémie Bouttier, Emmanuel Guitter +1 · 4 citations
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Black Holes and Theoretical Physics #Noncommutative and Quantum Gravity Theories #math-ph #math.CO #math.MP

paper · pdf · doi:10.1088/1742-5468/2008/07/p07020

published as J. Stat. Mech. (2008) P07020 · 43 pages, 16 color figures, misprints and figure 15 corrected

arxiv created 2008/07/24 · openalex publication_date 2008/07/24 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30

Abstract

We compute the generating function of random planar quadrangulations with three marked vertices at prescribed pairwise distances. In the scaling limit of large quadrangulations, this discrete three-point function converges to a simple universal scaling function, which is the continuous three-point function of pure 2D quantum gravity. We give explicit expressions for this universal three-point function in both the grand-canonical and canonical ensembles. Various limiting regimes are studied when some of the distances become large or small. By considering the case where the marked vertices are aligned, we also obtain the probability law for the number of geodesic points, namely vertices that lie on a geodesic path between two given vertices, and at prescribed distances from these vertices.

Citations

• The degree distribution in bipartite planar maps: applications to the Ising model
• Combinatorics of bicubic maps with hard particles

Cited by

• The Brownian map is the scaling limit of uniform random plane quadrangulations
• Distance statistics in large toroidal maps
• Scaling limits of random trees and planar maps
• Statistics of the Voronoï cell perimeter in large bi-pointed maps

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