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

• 成果类型: Article
• 署名作者: Claeys, T.; Kuijlaars, A. B. J.; Vanlessen, M.
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2008.168.601
• 发表日期: 2008
• 页码: 601-641
• 关键词: double scaling limit; riemann-hilbert problem; Orthogonal polynomials; strong asymptotics; UNIVERSALITY; solvability; spectrum; Respect; models; edge

摘要

We study unitary random matrix ensembles of the form Z(n,N)(-1)|det M|(2 alpha)e(-N Tr V(M)) dM, where alpha > -1/2 and V is such that the limiting mean eigenvalue density for n, N -> infinity 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 alpha)e(-NV(x)). Here the main focus is on the construction of a local parametrix near the origin with psi-functions associated with a special solution q(alpha), of the Painleve 11 equation q '' = sq + 2q(3) - alpha. We show that q(alpha) has no real poles for alpha > -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(alpha) in the double scaling limit.