Matrix models with hard walls: Geometry and solutions

TL;DR

This paper explores matrix models with hard walls, analyzing their solutions via advanced mathematical hierarchies, and introduces a diagrammatic technique to compute free energy in the large matrix limit.

Contribution

It extends the understanding of matrix models with confined eigenvalues by linking solutions to Riemann surfaces and developing a diagrammatic method for free energy calculation.

Findings

Solutions described by generalized Whitham--Krichever hierarchies.

Tau-functions derivatives relate to Riemann surfaces and WDVV equations.

Developed a diagrammatic technique for all-order free energy computation.

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.