Local Laws and a Mesoscopic CLT for $\beta$-ensembles

TL;DR

This paper establishes local laws and a mesoscopic central limit theorem for $eta$-ensembles, advancing understanding of fluctuations and energy at microscopic and mesoscopic scales in statistical mechanics.

Contribution

It introduces the first local laws for the log-gas energy and proves a mesoscopic CLT valid at all scales, extending previous results.

Findings

Control of local energy quantities at microscopic scales

Fluctuation control of linear statistics at all mesoscales

First proof of a CLT at arbitrary mesoscales for $eta$-ensembles

Abstract

We study the statistical mechanics of the log-gas, or $\beta$-ensemble, for general potential and inverse temperature. By means of a bootstrap procedure, we prove local laws on the next order energy that are valid down to microscopic length scales. To our knowledge, this is the first time that this kind of a local quantity has been controlled for the log-gas. Simultaneously, we exhibit a control on fluctuations of linear statistics that is valid at all mesoscales. Using these local laws, we are able to exhibit for the first time a CLT at arbitrary mesoscales, improving upon a previous result of Bekerman-Lodhia that was true only for power mesoscales.