q-rationals and dimers

2025/10/17 by Valentin Ovsienko, Ovsienko, Valentin
Mathematics · #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Random Matrices and Applications

paper · pdf · doi:10.48550/arxiv.2510.16270

openalex publication_date 2025/10/17 · openalex created_date 2025/10/22 · openalex updated_date 2026/07/28

Abstract

We describe the relationships between the notion of q-deformed rational numbers, introduced in our previous work with Sophie Morier-Genoud, and the theory of dimer models. We show that q-deformed rationals can be calculated in terms of perfect matchings of certain bipartite graphs, known as snake graphs, or ribbon tiles, etc. equipped with a certain weight function on the set of edges. We apply some elements of the dimer theory to get more information about q-rationals.

Citations

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• Positivity for cluster algebras from surfaces
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Related

• Dimers, filters, and q-deformed real numbers
• q-analogs of rational numbers: from Ostrowski numeration systems to perfect matchings
• q-deformed integers derived from pairs of coprime integers and its applications
• q-deformed rationals and irrationals
• q-Rationals and Finite Schubert Varieties