Large-$N$ Dynamics of a QCD-Inspired Unitary Matrix Model

We study the large-$N$ limit of $U(N)$ and $SU(N)$ unitary matrix models inspired by QCD. The model is analyzed in two cases: $\mu = 0$, where the potential is real, and finite $\mu$, where it becomes complex. The complex action drives the eigenvalues into the complex plane, leading to $\langle U \rangle \neq \langle U^{-1} \rangle$. In the ungapped phase, we obtain analytic expressions for the spectral density, Wilson loops, and free energy, which reproduce the low-temperature behaviour of QCD. In contrast, the gapped phase involves a nontrivial resolvent and is solved partially analytically and numerically. At $\mu=0$, the model exhibits a $3^{rd}$ order phase transition, while at finite $\mu$, it shows a continuous phase transition of at least second order.