  In [arXiv:1805.05057 [hep-th]],[arXiv:1812.00811 [hep-th]], the partition
function of the Gross-Witten-Wadia unitary matrix model with the logarithmic
term 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 that
contains 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 ODEs
for the scaling functions to the free energy, the time variable being the
scaled total mass and make a consistency check on the spectral curve on this
matrix model.
