**"Around
Large Random Matrices"**

By

** Abstract:** Random matrix theory is
roughly the study of the distribution of eigenvalues
of large random square matrix in which the distribution of the entries is
fixed. It features several mathematical structures : integrability,
topological recursion, relation to intersection theory on the moduli space of
curves, ... and conversely, the study of random matrices as a toy example can
help understand better those structures, their interplay and their
generalization. I will illustrate two of them, integrability
and topological recursion, while describing the behavior
of (real) eigenvalues in certain large random
matrices. If N -> infinity is the size of random matrix, the eigenvalues are globally distributed in a deterministic
way. The deviations to this behavior are
exponentially rare, and can be measured by concentration results. Then, the
moments of M can either have a 1/N asymptotic expansion, or feature an
oscillatory behavior at all orders in 1/N. In both
cases, the coefficient of the expansion can be computed by the
"topological recursion", which is a universal recursion involving algebraic
geometry on a curve. In our case, the curve is determined only by the leading
order of the mean and the covariance of eigenvalues. If time permits, I will explain how the
topological recursion allows to predict the tails of
the (universal) distribution of the fluctuations of the maximal eigenvalue near its mean in large random matrices in the beta
ensemble, known as the beta Tracy-Widom function.

**Date: ****Friday, September 13, 2013**

**Time: ****15.40 – 16:30 **

**Place: ****Mathematics Seminar Room,
SA-141**

Tea and cookies will be served before the
seminar.