We study the statistical mechanics of the log gas, an interacting particle system with applications in 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 are suitable for generalization to higher dimensional Riesz interactions; we will discuss some generalizations of the above approach and partial results for the Riesz gas in higher dimensions.