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

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.

💡 Research Summary

The paper investigates the singular sector of the Hermitian random matrix model (HRMM) in the large‑N limit under the zero‑genus (genus‑0) assumption. In this regime the model’s free‑energy functional reduces to a dispersionless integrable hierarchy, namely the dispersionless Toda (dToda) hierarchy. The authors first derive the hodograph equations for the dToda hierarchy, which express the endpoints of the eigenvalue support (often denoted u and v) as implicit functions of the external potentials (times). These equations can be written as a nonlinear system whose Jacobian matrix J determines regularity: det J ≠ 0 corresponds to smooth solutions, while det J = 0 signals the onset of a singularity (gradient catastrophe).

A central observation is that the condition det J = 0 is mathematically equivalent to the requirement that a solution of an Euler‑Poisson‑Darboux (EPD) equation of type E(a,a) with a = –½ be at a critical point. The EPD equation is a symmetric second‑order partial differential equation, \