We use the law of addition in random matrix theory to analyze the spectral distributions of a variety of chiral random matrix models as inspired from QCD whether through symmetries or models. In terms of the Blue's functions recently discussed by Zee, we show that most of the spectral distributions in the macroscopic limit and the quenched approximation, follow algebraically from the discontinuity of a pertinent solution to a cubic (Cardano) or a quartic (Ferrari) equation. We use the end-point equation of the energy spectra in chiral random matrix models to argue for novel phase structures, in which the Dirac density of states plays the role of an order parameter.
Nowak, M. A., Papp, G., & Zahed, I. (1996). QCD-inspired spectra from Blue’s functions. Physics Letters, Section B: Nuclear, Elementary Particle and High-Energy Physics, 389(1), 137–143. https://doi.org/10.1016/S0370-2693(96)01238-5