Optimal local laws and CLT for the circular Riesz gas

TL;DR

This paper establishes near-optimal local laws and a quantitative CLT for the one-dimensional Riesz gas on the circle, advancing understanding of particle interactions and fluctuations in long-range systems.

Contribution

It introduces new rigidity estimates and a CLT for linear statistics in the Riesz gas, handling very singular test functions through advanced analytical techniques.

Findings

Proved near-optimal gap rigidity estimates at all scales.

Established a quantitative CLT for linear statistics.

Handled very singular test functions using a comparison principle.

Abstract

We study the long-range one-dimensional Riesz gas on the circle, a continuous system of particles interacting through a Riesz kernel. We establish near-optimal rigidity estimates on gaps valid at any scale. Leveraging these local laws together with Stein's method, we prove a quantitative Central Limit Theorem for linear statistics. The proof is based on a mean-field transport and a fine analysis of the fluctuations of local error terms using various convexity and monotonicity arguments. Using a comparison principle for the Helffer-Sj\"ostrand equation, the method can handle very singular test functions, including characteristic functions of intervals.