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Partition functions for matrix models and isomonodromic tau functions. (English) Zbl 1050.37032
The goal of the paper is to investigate a generalization of the simplest version of the Hermitean one-matrix model, with measures that are exponentials of polynomial trace invariants, exactly what a relationship is. The main idea is to consider, for finite \(n\) (the size of the random matrix), the isomonodromic deformation systems satisfied by the associated sequence of orthogonal polynomials and to compute an explicit formula relating the associated isomonodromic tau function to the partition function of the generalized matrix model.

37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
82B23 Exactly solvable models; Bethe ansatz
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