We introduce the Beta-ensemble of random matrix theory and its so-called spectral measure. It is a measure on the unit circle that encodes all the ensemble. One of its distinctive features is that it exhibits a rich geometry. For instance, it has a non-trivial Hausdorff dimension. It turns out that many properties of this measure can be encoded in a sequence of coupled random walks in the plane. We will first show a toy model for these walks and some related results. We will then describe other results and conjectures on the spectral measure of the Beta-ensemble, and how it is related to the Gaussian Free Field and the Liouville Quantum Gravity. This is a joint work with Balint Virag (U. Toronto) and Tom Alberts (U. Utah).
Raoul Normand is visiting assistant professor at NYU Shanghai. Prior to joining NYU Shanghai, he was a postdoctoral fellow at the Academia Sinica in Taipei and at the University of Toronto. He holds a PhD from University Paris 6. His research interests include coagulation processes, particle systems, and random matrices.
Seminar by the NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai