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arXiv · 2509.24785

Discrete flat disks: rigid quadrangulations

Timothy Budd
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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-M\'elou 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.

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Timothy Budd. 2025-09-29. Discrete flat disks: rigid quadrangulations. https://arxiv.org/abs/2509.24785

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