Matrix models with hard walls: Geometry and solutions

Abstract

We discuss various aspects of most general multisupport solutions to matrix models in the presence of hard walls, i.e., in the case where the eigenvalue support is confined to subdomains of the real axis. The structure of the solution at the leading order is described by semiclassical, or generalized Whitham--Krichever hierarchies as in the unrestricted case. Derivatives of tau-functions for these solutions are associated with families of Riemann surfaces (with possible double points) and satisfy the Witten--Dijkgraaf--Verlinde--Verlinde equations. We then develop the diagrammatic technique for finding free energy of this model in all orders of the 't~Hooft expansion in the reciprocal matrix size generalizing the Feynman diagrammatic technique for the Hermitian one-matrix model due to Eynard.

46 pp., 2 figures in LaTeX; contribution to special volume of J.Phys.A on matrix models

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• A matrix model for the topological string II: The spectral curve and mirror geometry
• Spectral curves for hypergeometric Hurwitz numbers
• Complex (super)-matrix models with external sources and $q$-ensembles of Chern-Simons and ABJ(M) type
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• Matrix and vector models in the strong coupling limit