Multi-critical unitary random matrix ensembles and the general Painlevé II equation
(2008) Annals of Mathematics — Vol. 167, p. 601-642 (2008)
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- (en) We study unitary random matrix ensembles of the form $Z_{n,N}^{-1} |\det M|^{2\alpha} e^{-N \Tr V(M)}dM$, 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|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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Claeys, T., Kuijlaars, A., & Vanlessen, M. (2008). Multi-critical unitary random matrix ensembles and the general Painlevé II equation. Annals of Mathematics, 167, 601-642. https://doi.org/10.48550/arXiv.math-ph/0508062 (Original work published 2008)