{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/a-uniform-sampling-procedure-for-abstract","title":"A Uniform Sampling Procedure for Abstract Triangulations of Surfaces","arxiv_id":"2211.07798","date":"2022-11-14","proceeding":null,"authors":["Rajan Shankar","Jonathan Spreer"],"abstract":"We present a procedure to sample uniformly from the set of combinatorial isomorphism types of balanced triangulations of surfaces - also known as graph-encoded surfaces. For a given number $n$, the sample is a weighted set of graph-encoded surfaces with $2n$ triangles. The sampling procedure relies on connections between graph-encoded surfaces and permutations, and basic properties of the symmetric group. We implement our method and present a number of experimental findings based on the analysis of $138$ million runs of our sampling procedure, producing graph-encoded surfaces with up to $280$ triangles. Namely, we determine that, for $n$ fixed, the empirical mean genus $\\bar{g}(n)$ of our sample is very close to $\\bar{g}(n) = \\frac{n-1}{2} - (16.98n -110.61)^{1/4}$. Moreover, we present experimental evidence that the associated genus distribution more and more concentrates on a vanishing portion of all possible genera as $n$ tends to infinity. Finally, we observe from our data that the mean number of non-trivial symmetries of a uniformly chosen graph encoding of a surface decays to zero at a rate super-exponential in $n$.","url_abs":"https://arxiv.org/abs/2211.07798v1","url_pdf":"https://arxiv.org/pdf/2211.07798v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"a-uniform-sampling-procedure-for-abstract","repo_url":"https://github.com/rajan-shankar/uniform-sampling-procedure-for-triangulations","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}