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.