Optimal local laws and CLT for the circular Riesz gas
Probability
2025-07-22 v5 Mathematical Physics
math.MP
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.
Keywords
Cite
@article{arxiv.2112.05881,
title = {Optimal local laws and CLT for the circular Riesz gas},
author = {Jeanne Boursier},
journal= {arXiv preprint arXiv:2112.05881},
year = {2025}
}