We study the statistical mechanics of the log gas, an interacting particle system with applications to random matrix theory and statistical physics, for general potential and inverse temperature. By means of a bootstrap procedure, we prove local laws on a novel next order energy quantity that are valid down to microscopic length scales. Simultaneously, we exhibit a control on fluctuations of linear statistics that is also valid down to microscopic scales. Using these local laws, we exhibit for the first time a CLT at arbitrary mesoscales, improving upon previous results of Bekerman and Lodhia.
The methods we use generalize well to the study of higher dimensional Riesz gases; we will discuss some generalizations of the above approach and partial results on local laws and fluctuations for the Riesz gas in higher dimensions. This is joint work in progress with Sylvia Serfaty.