Discrete flat disks: rigid quadrangulations
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866911183786737664 |
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| author | Budd, Timothy |
| author_facet | Budd, Timothy |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_24785 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Discrete flat disks: rigid quadrangulations Budd, Timothy Combinatorics High Energy Physics - Theory Mathematical Physics Probability 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. |
| title | Discrete flat disks: rigid quadrangulations |
| topic | Combinatorics High Energy Physics - Theory Mathematical Physics Probability |
| url | https://arxiv.org/abs/2509.24785 |