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In [arXiv:1805.05057 [hep-th]],[arXiv:1812.00811 [hep-th]], the partitionfunction of the Gross-Witten-Wadia unitary matrix model with the logarithmicterm has been identified with the $\tau$ function of a certain Painlev\'{e}system, and the double scaling limit of the associated discrete Painlev\'{e}equation to the critical point provides us with the Painlev\'{e} II equation.This limit captures the critical behavior of the $su(2)$, $N_f =2$$\mathcal{N}=2$ supersymmetric gauge theory around its Argyres-Douglas $4D$superconformal point. Here, we consider further extension of the model thatcontains the $k$-th multicritical point and that is to be identified with$\hat{A}_{2k, 2k}$ theory. In the $k=2$ case, we derive a system of two ODEsfor the scaling functions to the free energy, the time variable being thescaled total mass and make a consistency check on the spectral curve on thismatrix model.

Multicritical points of unitary matrix model with logarithmic potential identified with Argyres-Douglas points