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.
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Research supported by the grant KAW 2010.0063 from the Knut and Alice Wallenberg Foundation.
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Communicated by Percy Deift and Alexander Its.
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Duits, M. Painlevé Kernels in Hermitian Matrix Models. Constr Approx 39, 173–196 (2014). https://doi.org/10.1007/s00365-013-9201-7
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DOI: https://doi.org/10.1007/s00365-013-9201-7
Keywords
- Hermitian matrix models
- Eigenvalue distribution
- Correlation kernel
- Critical phenomena
- Painlevé transcendents
- Biorthogonal polynomials
- Riemann–Hilbert problems