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

We study unitary random matrix ensembles of the form $Z_{n,N}^{-1} |\det M|^{2α} e^{-N \Tr V(M)}dM$, where $α>-1/2$ and $V$ is such that the limiting mean eigenvalue density for $n,N\to\infty$ and $n/N\to 1$ vanishes quadratically at the origin. In order to compute the double scaling limits of the eigenvalue correlation kernel near the origin, we use the Deift/Zhou steepest descent method applied to the Riemann-Hilbert problem for orthogonal polynomials on the real line with respect to the weight $|x|^{2α}e^{-NV(x)}$. Here the main focus is on the construction of a local parametrix near the origin with $ψ$-functions associated with a special solution $q_α$ of the Painlevé II equation $q''=sq+2q^3-α$. We show that $q_α$ has no real poles for $α> -1/2$, by proving the solvability of the corresponding Riemann-Hilbert problem. We also show that the asymptotics of the recurrence coefficients of the orthogonal polynomials can be expressed in terms of $q_α$ in the double scaling limit.