Two-matrix model with semiclassical potentials and extended Whitham hierarchy

TL;DR

This paper analyzes a two-matrix model with rational potentials and hard-edge spectra, deriving explicit formulas for free energy, correlation functions, and extending Whitham hierarchy calculus with new residue formulae.

Contribution

It provides explicit formulas for the planar free energy and correlation functions, extending classical variational calculus and Whitham hierarchy to include additional data.

Findings

Explicit formula for the planar free energy.

Worked out four-point correlation functions.

Extended Whitham hierarchy formalism with residue formulas.

Abstract

We consider the two-matrix model with potentials whose derivative are arbitrary rational function of fixed pole structure and the support of the spectra of the matrices are union of intervals (hard-edges). We derive an explicit formula for the planar limit of the free energy and we derive a calculus which allows to compute derivatives of arbitrarily high order by extending classical Rauch's variational formulae. The four-points correlation functions are explicitly worked out. The formalism extends naturally to the computation of residue formulae for the tau function of the so-called universal Whitham hierarchy studied mainly by I. Krichever: our setting extends that moduli space in that there are certain extra data.