Chiral Random Matrix Models : Thermodynamics, Phase Transitions and Universality

For one flavour, we observe that standard chiral random matrix models as inspired from QCD spin and color symmetry, are a simplified version of the Nambu-Jona-Lasinio (NJL) model whether in vacuum or matter. The ensuing thermodynamics is that of constituent quarks, with an entropy driven second-order transition and mean-field exponents. At zero temperature but finite chemical potential $\mu$, the transition is first order, with a critical $\mu$ of the order of the constituent quark mass. The transition occurs at zero baryon density, since chiral random matrix models do not support a Fermi surface. For two and three flavours non-standard chiral random matrix models with $U_A(1)$ breaking are suggested, with mean-field phase transitions in large $N$. For three flavours and small current masses (weak field), the chiral random matrix model exhibits a first order transition remiscent of the isotropic-nematic transition in liquid crystals. The latter displays a tricritical point for strong fields. Some issues related to confinement are briefly discussed.