Modul: MAT075 Zurich Graduate Colloquium

## What is... a random matrix?

Johannes Alt talk

**Date:** 05.03.19 **Time:** 17.15 - 18.30 **Room:**

Random matrices were introduced in the first half of the 20th century by Wishart and Wigner for applications in statistics and quantum physics, respectively. As of today, many other applications of random matrices, in physics, biology and engineering, as well as many connections to other branches of mathematics have been discovered.

In this talk, we give an introduction to certain aspects of the mathematical theory of random matrices. Moreover, we present some recent results on the eigenvalue density of large Hermitian random matrices with correlated entries and general expectation. Typically, their eigenvalue density becomes deterministic when the matrix size goes to infinity and this limit is determined by the Matrix Dyson equation. Under general conditions on the correlations and expectation, the limiting eigenvalue density is real analytic apart from finitely many square root edges and cubic root cusps. Close to a square root edge the fluctuation of the eigenvalues is governed by the famous Tracy-Widom distribution that only depends on the basic symmetry type of the random matrix. This is joint work with László Erdős, Torben Krüger and Dominik Schröder.