The resolvent of many prominent random matrix ensembles can approximately be described by the fundamental Dyson equation involving only first and second moments. Historically, Dyson equation was used to describe the density of states on global scales. Recent advances toward generalisations of the Wigner-Dyson-Mehta universality conjecture require to understand the spectrum on local scales that reveal another mathematical aspects of the Dyson equation.

These talks will focus on the role of the Dyson equation in proving {local laws}, i.e.statements on the eigenvalue density on scales sightly above the typical eigenvalue spacing. We also demonstrate how such local laws help prove the universality of eigenvalue correlations.

Please see the lecture slides here.

Instructor

Professor Laszlo Erdös (http://pub.ist.ac.at/~lerdos/ and https://ist.ac.at/research/mathematics-and-computer-science/erdoes-group/) from the Institute of Science and Technology Austria (IST Austria).