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We continue to study the matrix model of the $N_f =2$ $SU(2)$ case thatrepresents the irregular conformal block. What provides us with thePainlev\'{e} system is not the instanton partition function per se but rather afinite analog of its Fourier transform that can serve as a generating function.The system reduces to the extension of the Gross-Witten-Wadia unitaryone-matrix model by the logarithmic potential while keeping the planar criticalbehavior intact. The double scaling limit to this critical point is aconstructive way to study Argyres-Douglas type theory from IR. We elaborateupon the method of orthogonal polynomial and its relevance to these problems,developing it further for the case of a generic unitary matrix model and thatof a special case with the logarithmic potential.

Discrete Painleve system for the partition function of $N_f =2$ $SU(2)$ supersymmetric gauge theory and its double scaling limit