Random matrix theory and the sixth Painlevé equation
نویسندگان
چکیده
منابع مشابه
Dynamics of the Sixth Painlevé Equation
— The sixth Painlevé equation is hiding extremely rich geometric structures behind its outward appearance. In this article, we give a complete picture of its dynamical nature based on the Riemann-Hilbert approach recently developed by the authors and using various techniques from algebraic geometry. A large part of the contents can be extended to Garnier systems, while this article is restricte...
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A feature of certain ensembles of random matrices is that the corresponding measure is invariant under conjugation by unitary matrices. Study of such ensembles realised by matrices with Gaussian entries leads to statistical quantities related to the eigenspectrum, such as the distribution of the largest eigenvalue, which can be expressed as multidimensional integrals or equivalently as determin...
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We consider two Lax systems for the homogeneous Painlevé II equation: one of size 2×2 studied by Flaschka and Newell in the early 1980s, and one of size 4×4 introduced by Delvaux, Kuijlaars, and Zhang and Duits and Geudens in the early 2010s. We prove that solutions to the 4×4 system can be derived from those to the 2 × 2 system via an integral transform, and consequently relate the Stokes mult...
متن کاملDiscrete Painlevé Equations and Random Matrix Averages
The τ-function theory of Painlevé systems is used to derive recurrences in the rank n of certain random matrix averages over U (n). These recurrences involve auxilary quantities which satisfy discrete Painlevé equations. The random matrix averages include cases which can be interpreted as eigenvalue distributions at the hard edge and in the bulk of matrix ensembles with unitary symmetry. The re...
متن کاملOn the asymptotics of the real solutions to the general sixth Painlevé equation
The mathematical and physical significance of the six Painlevé transcendents has been well established. In the last 20 to 30 years, many mathematicians have spent dramatic effort on studying the properties of these transcendents. Although it is the most complicated one among the six Painlevé equations, there have been many results about the sixth Painlevé transcendent. In fact, the asymptotics ...
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ژورنال
عنوان ژورنال: Journal of Physics A: Mathematical and General
سال: 2006
ISSN: 0305-4470,1361-6447
DOI: 10.1088/0305-4470/39/39/s14