Large N expansions and Painlevé hierarchies in the Hermitian matrix model

Abstract

We present a method to characterize and compute the large N formal asymptotics of regular and critical Hermitian matrix models with general even potentials in the one-cut and two-cut cases. Our analysis is based on a method to solve continuum limits of the discrete string equation which uses the resolvent of the Lax operator of the underlying Toda hierarchy. This method also leads to an explicit formulation, in terms of coupling constants and critical parameters, of the members of the Painlevé I and Painlevé II hierarchies associated with one-cut and two-cut critical models respectively.

References in corpus (5)

• Universal distribution of random matrix eigenvalues near the "birth of a cut" transition
• Large N expansion of convergent matrix integrals, holomorphic anomalies, and background independence
• The double scaling limit method in the Toda hierarchy
• Phase transitions in multi-cut matrix models and matched solutions of Whitham hierarchies
• Semiclassical expansions in the Toda hierarchy and the hermitian matrix model