A formal framework for higher-order spin models via hypergraphs, polymatroids, and the Tutte polynomial

[2608.14628] A formal framework for higher-order spin models via hypergraphs, polymatroids, and the Tutte polynomial Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Combinatorics arXiv:2608.14628 (math) [Submitted on 23 Jul 2026] Title: A formal framework for higher-order spin models via hypergraphs, polymatroids, and the Tutte polynomial Authors: Khallil Berrekkal , Joanna A. Ellis-Monaghan , Merijn Moody , Clélia de Mulatier View a PDF of the paper titled A formal framework for higher-order spin models via hypergraphs, polymatroids, and the Tutte polynomial, by Khallil Berrekkal and 3 other authors View PDF HTML (experimental) Abstract: We develop a rigorous mathematical framework for statistical mechanics models on hypergraphs, and give conditions for lifting the classical connection between Potts model partition functions and the Tutte polynomial from graphs to hypergraphs. We define hypergraphical models and their associated partition functions, and extend these to special classes of models induced by families of interaction functions. For boolean interaction families, whose interaction functions map to $\{0,1\}$, we show that the partition function is determined by a combinatorial rank function, and we establish sufficient conditions for a hypergraph deletion-contraction recurrence and for when the rank function defines a polymatroid. We illustrate the theory by applying it to three hypergraph interaction families: Parity Ising, Delta Potts, and And Ising. The induced hypergraphical models are not isomorphic to each other, but the first two reduce to the same graphical Ising models. We identify three polymatroids naturally associated with hypergraphs for these models: respectively, the binary matroid of the incidence matrix over $\mathbb{F}_2$, the hypergraphical polymatroid, and the boolean polymatroid. For graphs, the first two reduce to the classical graphical matroid, and their partition functions recover the multivariate Tutte polynomial. The Tutte polynomial thus admits at least two distinct generalizations for hypergraphs, both satisfying a deletion-contraction recurrence: the Tutte polynomial of the binary matroid of the hypergraph incidence matrix, and a multivariate version of the Poincaré polynomial of the hypergraphical polymatroid. The partition functions of And Ising models are likewise multivariate versions of the Poincaré polynomial, here of the boolean polymatroid. These examples illustrate the much greater range of hypergraphical models and underscore the need for the unifying theory. Subjects: Combinatorics (math.CO) ; Statistical Mechanics (cond-mat.stat-mech); Mathematical Physics (math-ph) MSC classes: 82B20, 05C31, 05C65, 05B35 Cite as: arXiv:2608.14628 [math.CO] (or arXiv:2608.14628v1 [math.CO] for this version) https://doi.org/10.48550/arXiv.2608.14628 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Merijn Moody [ view email ] [v1] Thu, 23 Jul 2026 18:29:52 UTC (58 KB) Full-text links: Access Paper: View a PDF of the paper titled A formal framework for higher-order spin models via hypergraphs, polymatroids, and the Tutte polynomial, by Khallil Berrekkal and 3 other authors View PDF HTML (experimental) TeX Source view license Current browse context: math.CO < prev | next > new | recent | 2026-08 Change to browse by: cond-mat cond-mat.stat-mech math math-ph math.MP References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... 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