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arXiv · 2112.05881

Optimal local laws and CLT for the circular Riesz gas

Jeanne Boursier
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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.

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Jeanne Boursier. 2021-12-11. Optimal local laws and CLT for the circular Riesz gas. https://arxiv.org/abs/2112.05881

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