Resolvents and Seiberg-Witten representation for Gaussian beta-ensemble

TL;DR

This paper demonstrates that the Seiberg-Witten equations and resolvent properties of matrix models extend to Gaussian beta-ensembles, using recursion methods to analyze their genus expansion despite the lack of integrability tools.

Contribution

It establishes the preservation of Seiberg-Witten properties in beta-ensembles and applies recursion techniques to analyze the Gaussian case.

Findings

Seiberg-Witten equations hold for beta-ensembles.

Exact resolvent properties are maintained in the generalization.

Recursion methods enable genus expansion analysis for Gaussian beta-ensembles.

Abstract

The exact free energy of matrix model always obeys the Seiberg-Witten (SW) equations on a complex curve defined by singularities of the quasiclassical resolvent. The role of SW differential is played by the exact one-point resolvent. We show that these properties are preserved in generalization of matrix models to beta-ensembles. However, since the integrability and Harer-Zagier topological recursion are still unavailable for beta-ensembles, we need to rely upon the ordinary AMM/EO recursion to evaluate the first terms of the genus expansion. Consideration in this paper is restricted to the Gaussian model.