  The generating functions for the gauge theory observables are often
represented in terms of the unitary matrix integrals. In this work, the
perturbative and non-perturbative aspects of the generic multi-critical unitary
matrix models are studied by adopting the integrable operator formalism, and
the multi-critical generalization of the Tracy--Widom distribution in the
context of random partitions. We obtain the universal results for the
multi-critical model in the weak and strong coupling phases. The free energy of
the instanton sector in the weak coupling regime, and the genus expansion of
the free energy in the strong coupling regime are explicitly computed and the
universal multi-critical phase structure of the model is explored. Finally, we
apply our results in concrete examples of supersymmetric indices of gauge
theories in the large $N$ limit.
