Luke Peilen (Temple): Local laws and fluctuations for super-Coulombic Riesz gases

Coulomb and Riesz gases are interacting particle systems with a wide range of applications in random matrix theory, approximation theory, convex geometry, and diverse areas of physics. We study the statistical mechanics of general Riesz gases at mesoscopic and microscopic length scales, providing controls on fluctuations of linear statistics down to microscopic length scales and establishing for the first time a CLT for fluctuations of linear statistics for general two-dimensional Riesz gases.

A novel technical difficulty involves the development of a transport method for general Riesz gases, building on work of Leblé and Serfaty for Coulomb gases, to understand the behavior of the partition function under small perturbations of the external potential. Our study involves several questions concerning degenerate, singular elliptic PDE and fractional operators.

This is based on joint work with S. Serfaty.