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We study the partition function of the matrix model of finite size thatrealizes 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 obtainedin [arXiv:1008.1861 [hep-th]] as the massive scaling limit of the $\beta$deformed matrix model representing the conformal block. We point out that themodel for the case of $\beta =1$ can be recast into a unitary matrix model withlog potential and show that it is exhibited as a discrete Painlev\'{e} systemby the method of orthogonal polynomials. We derive the Painlev\'{e} IIequation, taking the double scaling limit in the vicinity of the critical pointwhich is the Argyres-Douglas type point of the corresponding spectral curve. Bythe $0$d-$4$d dictionary, we obtain the time variable and the parameter of thedouble scaled theory respectively from the sum and the difference of the twomass parameters scaled to their critical values.

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