Student Probability Seminar

In this review talk, we will introduce the history and classical theory of the log gas, or beta ensemble, on the real line. One such classical result is that the empirical measure of the particles converges to a deterministic equilibrium measure. In order to understand this convergence more, we will look at integrals of smooth test functions against the difference of these measures, or, fluctuations of linear statistics. Under suitable assumptions, these fluctuations are asymptotically Gaussian. We will consider the result of Bekerman-Leble-Serfaty (2018), which makes use of the pioneering method of Johansson (1998) to understand such fluctuations, and the electric approach introduced in Sandier-Serfaty (2015) for analyzing Coulomb and Riesz interactions. Time permitting, we will discuss extensions of these results to mesoscopic scales.