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Partial differential equations in random matrix theory

Banff International Research Station for Mathematical Innovation and Discovery

I will explain how tools from the theory of partial differential equations can be used to compute the eigenvalue distribution of large random matrices. I will discuss several examples where this method can be used and show lots of pictures illustrating the results. I will then explain how the method works in the simplest interesting example, for random matrices of the form $X+iY$, where $X$ is drawn from the Gaussian Unitary Ensemble and $Y$ is an arbitrary Hermitian random matrix independent of $X$. The talk should be accessible to a wide audience. The PDE approach to the subject was introduced in a work of mine with Bruce Driver and Todd Kemp and the specific example I will discuss is joint work of mine with Ching Wei Ho.

Mathematics; Probability Theory And Stochastic Processes; Global Analysis; Analysis On Manifolds

video/mp4

Author affiliation: University of Notre Dame

2021-09-06

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/79657

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/