# Multi-critical unitary random matrix ensembles and the general Painlevé II equation | Annals of Mathematics

Abstract We study unitary random matrix ensembles of the form Z−1n,N|detM|2αe−NTrV(M)dM, where α>−1/2 and V is such that the limiting mean eigenvalue density for n,N→∞ and n/N→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|2αe−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 qα of the Painlevé II equation q”=sq+2q3−α. We show that qα 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 qα in the double scaling limit.

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### 数学年刊（Annals of Mathematics）

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