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We study unitary random matrix ensembles of the form $Z_{n,N}^{-1} |\detM|^{2\alpha} e^{-N \Tr V(M)}dM$, where $\alpha>-1/2$ and $V$ is such that thelimiting mean eigenvalue density for $n,N\to\infty$ and $n/N\to 1$ vanishesquadratically at the origin. In order to compute the double scaling limits ofthe eigenvalue correlation kernel near the origin, we use the Deift/Zhousteepest descent method applied to the Riemann-Hilbert problem for orthogonalpolynomials on the real line with respect to the weight$|x|^{2\alpha}e^{-NV(x)}$. Here the main focus is on the construction of alocal parametrix near the origin with $\psi$-functions associated with aspecial solution $q_\alpha$ of the Painlev\'e II equation $q''=sq+2q^3-\alpha$.We show that $q_\alpha$ has no real poles for $\alpha > -1/2$, by proving thesolvability of the corresponding Riemann-Hilbert problem. We also show that theasymptotics of the recurrence coefficients of the orthogonal polynomials can beexpressed in terms of $q_\alpha$ in the double scaling limit.

Multi-critical unitary random matrix ensembles and the general Painlevé II equation