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We study unitary random matrix ensembles of the form −1 Zn,N | 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 → ∞ 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|2α e−N V (x) . . | Annals of Mathematics Multi-critical unitary random matrix ensembles and the general Painlev_e II equation By T. Claeys A.B.J. Kuijlaars and M. Vanlessen Annals of Mathematics 167 2008 601 641 Multi-critical unitary random matrix ensembles and the general Painlevé II equation By T. CLAEys A.B.J. Kuijlaars and M. Vanlessen Abstract We study unitary random matrix ensembles of the form Z NI det MI2 e-N V M lM where a 1 2 and V is such that the limiting mean eigenvalue density for n N 1 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 2 e-NV x . Here the main focus is on the construction of a local parametrix near the origin with -functions associated with a special solution qa of the Painlevé II equation q sq 2q3 a. We show that qa has no real poles for a 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 qa in the double scaling limit. 1. Introduction and statement of results 1.1. Unitary random matrix ensembles. For n 2 N N 0 and a 1 2 we consider the unitary random matrix ensemble 1.1 Z NI det MI2 e-NTV M dM on the space of n X n Hermitian matrices M where V R R is a real analytic function satisfying 1.2 V x lim Ĩ . Z 1 log x2 1 Because of 1.2 and a 1 2 the integral 1.3 ZnN Ị I det MI2 e-NTrV M dM 602 T. CLAEYS A.B.J. KUIJLAARS AND M. VANLESSEN converges and the matrix ensemble 1.1 is well- defined. It is well known see for example 11 36 that the eigenvalues of M are distributed according to a determinantal point process with a correlation kernel given by n 1 1.4 Kn N x y x e v x y e v y XPk N x Pk N y k o where pk N Kk Nxk Kk N 0 denotes the k-th degree orthonormal
