Local Laws and Fluctuations for Super-Coulombic Riesz Gases

TL;DR

This paper establishes local laws and fluctuation results for super-Coulombic Riesz gases, demonstrating convergence to fractional Gaussian fields and addressing challenges from the nonlocal Riesz kernel.

Contribution

It introduces a bootstrap method to prove local laws and fluctuations for Riesz gases, extending techniques to nonlocal interactions and deriving a CLT for small inverse powers.

Findings

Proved local laws down to microscopic scales.

Established a CLT for Riesz gases with small inverse powers.

Showed convergence to fractional Gaussian fields.

Abstract

We study the local statistical behavior of the super-Coulombic Riesz gas of particles in Euclidean space of arbitrary dimension, with inverse power distance repulsion integrable near $0$, and with a general confinement potential, in a certain regime of inverse temperature. Using a bootstrap procedure, we prove local laws on the next order energy and control on fluctuations of linear statistics that are valid down to the microscopic lengthscale, and provide controls for instance, on the number of particles in a (mesoscopic or microscopic) box, and the existence of a limit point process up to subsequences. As a consequence of the local laws, we derive an almost additivity of the free energy that allows us to exhibit for the first time a CLT for Riesz gases corresponding to small enough inverse powers, at small mesoscopic length scales, which can be interpreted as the convergence of the associated potential to a fractional Gaussian field. Compared to the Coulomb interaction case, the main new issues arise from the nonlocal aspect of the Riesz kernel. This manifests in (i) a novel technical difficulty in generalizing the transport approach of Lebl\'e and the second author to the Riesz gas which now requires analyzing a degenerate and singular elliptic PDE, (ii) the fact that the transport map is not localized, which makes it more delicate to localize the estimates, (iii) the need for coupling the local laws and the fluctuations control inside the same bootstrap procedure.