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Description

Random matrix theory connects questions about eigenvalue distributions to problems in analysis and combinatorics. Orthogonal polynomials play a central role in these connections, while their characterization through Riemann-Hilbert problems provides a powerful framework for asymptotic analysis. I will begin by introducing these ideas and explaining how the large-matrix-size expansion of the free energy encodes the enumeration of maps, which are graphs embedded on surfaces with topological disks as complementary regions. In this expansion, successive terms correspond to surfaces of increasing genus. I will then discuss recent results on counting maps with even vertex degrees. For maps whose vertices all have the same degree, bivariate explicit enumeration formulas valid for arbitrary degree and number of vertices were previously known in genera zero and one. We obtain such formulas in genera two, three, and four, together with asymptotics as the degree tends to infinity. Our methods extend, in principle, to any fixed genus, with increasing computational effort. For maps with vertices of different even degrees, we recover Tutte’s 1962 formula for genus zero, originally obtained by purely combinatorial methods, and derive an exact enumeration formula in genus one. We also obtain universal asymptotics in arbitrary fixed genus as the number of vertices grows. These results combine orthogonal polynomial methods with analytic combinatorics in several variables and reveal a connection between the asymptotic counting constants and the Painlevé I equation. The talk is based on joint work with Tomas Lasic Latimer and with Ahmad Barhoumi and Nathan Hayford.