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Constructive Approximation

, Volume 39, Issue 1, pp 173–196 | Cite as

Painlevé Kernels in Hermitian Matrix Models

  • Maurice Duits
Article

Abstract

After reviewing the Hermitian one-matrix model, we will give a brief introduction to the Hermitian two-matrix model and present a summary of some recent results on the asymptotic behavior of the two-matrix model with a quartic potential. In particular, we will discuss a limiting kernel in the quartic/quadratic case that is constructed out of a 4×4 Riemann–Hilbert problem related to the Painlevé II equation. Also an open problem will be presented.

Keywords

Hermitian matrix models Eigenvalue distribution Correlation kernel Critical phenomena Painlevé transcendents Biorthogonal polynomials Riemann–Hilbert problems 

Mathematics Subject Classification

30E25 60B20 82B26 15B52 30F10 31A05 42C05 

Notes

Acknowledgements

Research supported by the grant KAW 2010.0063 from the Knut and Alice Wallenberg Foundation.

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of MathematicsRoyal Institute of Technology (KTH)StockholmSweden

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