Defranco, Mario; Gunnells, Paul E. Hypergraph matrix models. (English) Zbl 1499.81077 Mosc. Math. J. 21, No. 4, 737-766 (2021). Summary: The classical GUE matrix model of \(N\times N\) Hermitian matrices equipped with the Gaussian measure can be used to count the orientable topological surfaces by genus obtained through gluing the edges of a polygon. We introduce a variation of the GUE matrix model that enumerates certain edge-ramified CW complexes obtained from polygon gluings. We do this by replacing the Gaussian measure with a formal analogue related to generating functions that enumerate uniform hypergraphs. Our main results are three different ways to compute expectations of traces of powers. In particular, we show that our matrix model has a topological expansion. Cited in 1 Document MSC: 81T18 Feynman diagrams 16W10 Rings with involution; Lie, Jordan and other nonassociative structures Keywords:matrix models; hypergraphs; hyperbag graphs Software:nauty PDFBibTeX XMLCite \textit{M. Defranco} and \textit{P. E. Gunnells}, Mosc. Math. J. 21, No. 4, 737--766 (2021; Zbl 1499.81077) Full Text: Link References: [1] C. Berge, Graphs and hypergraphs, North-Holland Publishing Co., Amsterdam-London; · Zbl 0483.05029 [2] American Elsevier Publishing Co., Inc., New York, 1973. MR 0357172 [3] P. Etingof, Mathematical ideas and notions of quantum field theory, available from www-math. mit.edu/ etingof/, 2002. [4] B. Eynard, Topological expansion for the 1-Hermitian matrix model correlation functions, J. High Energy Phys. (2004), no. 11, 031, 35 pp. (2005). MR 2118807 [5] B. Eynard, Counting surfaces, Progress in Mathematical Physics, vol. 70, Birkhäuser/ Springer, [Cham], 2016. MR 3468847. CRM Aisenstadt chair lectures. · Zbl 1338.81005 [6] B. Eynard and N. Orantin, Invariants of algebraic curves and topological expansion, Com-mun. Number Theory Phys. 1 (2007), no. 2, 347-452. MR 2346575 · Zbl 1161.14026 [7] P. E. Gunnells, Generalized Catalan numbers from hypergraphs, submitted. · Zbl 1459.05006 [8] P. E. Gunnells, Exotic matrix models: the Albert algebra and the spin factor, Mosc. Math. J. 18 (2018), no. 2, 321-347. MR 3831011 · Zbl 1417.81153 [9] J. Harer and D. Zagier, The Euler characteristic of the moduli space of curves, Invent. Math. 85 (1986), no. 3, 457-485. MR 848681 · Zbl 0616.14017 [10] P. Jordan, J. von Neumann, and E. Wigner, On an algebraic generalization of the quantum mechanical formalism, Ann. of Math. (2) 35 (1934), no. 1, 29-64. MR 1503141 · JFM 60.0902.02 [11] M. Koecher, The Minnesota notes on Jordan algebras and their applications, Lecture Notes in Mathematics, vol. 1710, Springer-Verlag, Berlin, 1999. MR 1718170 · Zbl 1072.17513 [12] S. K. Lando and A. K. Zvonkin, Graphs on surfaces and their applications, Encyclopaedia of Mathematical Sciences, vol. 141, Springer-Verlag, Berlin, 2004. MR 2036721 · Zbl 1040.05001 [13] B. McKay, Nauty, available at http://pallini.di.uniroma1.it. [14] M. Mulase and A. Waldron, Duality of orthogonal and symplectic matrix integrals and quater-nionic Feynman graphs, Comm. Math. Phys. 240 (2003), no. 3, 553-586. MR 2005857 · Zbl 1033.81062 [15] X. Ouvrard, J. Le Goff, and S. Marchand-Maillet, Adjacency and tensor representation in general hypergraphs. Part 2: Multisets, hb-graphs and related e-adjacency tensors, preprint arXiv:1805.11952 [cs.DM] · Zbl 1460.05136 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.