Claeys, Tom; Kuijlaars, Arno B. J.; Vanlessen, Maarten Multi-critical unitary random matrix ensembles and the general Painlevé II equation. (English) Zbl 1179.15037 Ann. Math. (2) 168, No. 2, 601-641 (2008). The authors study unitary random matrix ensembles. In order to compute the double scaling limits of the eigenvalue correlation kernel near the origin, they use the P. Deift and X. Zhou [Ann. Math. (2) 137, No. 2, 295–368 (1993; Zbl 0771.35042)] steepest descent method applied to the Riemann-Hilbert (RH) problem for orthogonal polynomials on the real line. The paper is divided into the sections: Introduction and statement of results; The RH problem for Painlevé II and proof of Theorem 1.1; Steepest analysis of the RH problem; Proof of Theorm 1.2 and Proof of Theorem 1.7. Reviewer: Yueh-er Kuo (Knoxville) Cited in 1 ReviewCited in 45 Documents MSC: 15B52 Random matrices (algebraic aspects) 31A25 Boundary value and inverse problems for harmonic functions in two dimensions 35Q15 Riemann-Hilbert problems in context of PDEs 82B23 Exactly solvable models; Bethe ansatz Keywords:unitary Random matrix; steepest descent method; orthogonal polynomials; Riemann-Hilbert problem Citations:Zbl 0771.35042 × Cite Format Result Cite Review PDF Full Text: DOI arXiv