Discrete Painleve system and the double scaling limit of the matrix model for irregular conformal block and gauge theory

We study the partition function of the matrix model of finite size that realizes the irregular conformal block for the case of the ${\cal N}=2$ supersymmetric $SU(2)$ gauge theory with $N_f =2$. This model has been obtained in [arXiv:1008.1861 [hep-th]] as the massive scaling limit of the $\beta$ deformed matrix model representing the conformal block. We point out that the model for the case of $\beta =1$ can be recast into a unitary matrix model with log potential and show that it is exhibited as a discrete Painlev\'{e} system by the method of orthogonal polynomials. We derive the Painlev\'{e} II equation, taking the double scaling limit in the vicinity of the critical point which is the Argyres-Douglas type point of the corresponding spectral curve. By the $0$d-$4$d dictionary, we obtain the time variable and the parameter of the double scaled theory respectively from the sum and the difference of the two mass parameters scaled to their critical values.