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

2005/08/31 by T. Claeys, A. B. J. Kuijlaars, M. Vanlessen · 1 citation
Physics and Astronomy · Mathematics · #math-ph #math.CV #math.MP #nlin.SI #msc:15A52 #msc:31A25 #msc:35Q15 #msc:82B23

paper · pdf

published as Annals of Mathematics 168 (2008), 601--642 · 37 pages, 4 figures

arxiv created 2005/08/31 · arxiv updated 2010/07/30

Abstract

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→∞ 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 q_α of the Painlevé II equation q''=sq+2q³-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.

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