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MATHEMATICS COLLOQUIUM
"Universality in random matrix theory"
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Date: |
Download-files: |
Time: |
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Wednesday, 22. Feb 2017 |
Video-Recording for any system with MP4-support
- Video.mp4 (ca.391 Mb) |
15:15 – 16:20 |
Arno Kuijlaars
(Katholieke
Abstract :
Eigenvalues of
large random matrices have a remarkable behavior as the size of the
matrices tends to
infinity. The local repulsion of eigenvalues leads to microscopic
limit laws that
are independent of the fine details of the random matrix model.
This universality
phenomenon was observed by Wigner in the 1950s and proved by
mathematicians in
great generality since the 1990s.
Deviations from
universality correspond to phase transitions in limiting eigenvalue
behavior that can
be analyzed with nonlinear special functions: the Painlevé transcendents.
For example the
breaking of a spectral gap corresponds to a special solution of the
Painlevé II
equation.
In the talk I
will give an overview of these developments. I will also discuss the more
recently
discovered universal behavior for eigenvalues and singular values of products
of random
matrices.