Discrete flat disks: rigid quadrangulations

2025/09/29 by Budd, Timothy · 1 citation
#Combinatorics (math.CO) #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Mathematical Physics (math-ph) #Probability (math.PR)

paper · doi:10.48550/arxiv.2509.24785

Abstract

Inspired by a question of Ferrari in the physics context of JT gravity, we introduce and enumerate a combinatorial family of quadrangulations of the disk, called rigid quadrangulations. These form a subclass of the flat quadrangulations in the sense that every inner vertex has degree 4, and therefore it can be viewed as a discrete model of flat metrics on the disk. Our main result is a bijection between rigid quadrangulations and certain colorful integer-labeled quadrangulations of the sphere, together with a dictionary relating a variety of natural statistics on both sides. Adaptions of the bijection to various boundary conditions allow us to import recent enumerative results for colorful quadrangulation obtained by Bousquet-Mélou and Elvey Price. We discuss some consequences of the enumeration of rigid quadrangulations for a flat version of JT gravity at finite cutoff, and comment on potential scaling limits.

Citations

• A tree bijection for rigid quadrangulations
• Refined enumeration of planar Eulerian orientations
• Gaskets of O(2) loop-decorated random planar maps
• Self-similar Markov trees and scaling limits
• Random Disks of Constant Curvature: the Lattice Story
• A bijection between rooted planar maps and generalized fighting fish
• A bijection between Tamari intervals and extended fighting fish
• Mating of trees for critical Liouville quantum gravity
• The six-vertex model on random planar maps revisited
• Growth-fragmentation process embedded in a planar Brownian excursion
• Combinatorial Properties of Self-Overlapping Curves and Interior Boundaries
• Mating of trees for random planar maps and Liouville quantum gravity: a survey
• JT Gravity and the Ensembles of Random Matrix Theory
• JT gravity as a matrix integral
• The peeling process on random planar maps coupled to an O(n) loop model (with an appendix by Linxiao Chen)
• Fighting Fish: enumerative properties
• Martingales in self-similar growth-fragmentations and their connections\n with random planar maps
• Quantum gravity and inventory accumulation

Cited by

• A tree bijection for rigid quadrangulations

Related

• A tree bijection for rigid quadrangulations
• Random Disks of Constant Curvature: the Lattice Story
• Discrete Approaches Towards the Definition of a Quantum Theory of\n Gravity
• From Asymptotically Flat Gravity to Finite Causal Diamonds
• Piecewise Flat Metrics and Quantum Gravity