On the singular sector of the Hermitian random matrix model in the large N limit

TL;DR

This paper investigates the singular sector of the Hermitian random matrix model at large N, revealing deep connections between the dispersionless Toda hierarchy and the Benney hierarchy through Euler-Poisson-Darboux equations.

Contribution

It establishes a novel link between the singular sectors of two integrable hierarchies via critical points of Euler-Poisson-Darboux equations.

Findings

Identifies the singular sector of the zero genus case in the Hermitian random matrix model.

Shows the deep connection between the hodograph solutions of the dToda and Benney hierarchies.

Links the critical points of solutions of Euler-Poisson-Darboux equations to the singular sectors.

Abstract

The singular sector of zero genus case for the Hermitian random matrix model in the large N limit is analyzed. It is proved that the singular sector of the hodograph solutions for the underlying dispersionless Toda hierarchy and the singular sector of the 1-layer Benney (classical long wave equation) hierarchy are deeply connected. This property is due to the fact that the hodograph equations for both hierarchies describe the critical points of solutions of Euler-Poisson-Darboux equations E(a,a), with a=-1/2 for the dToda hierarchy and a=1/2 for the 1-layer Benney hierarchy.