Local Laws and Fluctuations for Super-Coulombic Riesz Gases
Luke Peilen, Sylvia Serfaty
TL;DR
The paper develops a comprehensive framework to study local laws and fluctuations for super-Coulombic Riesz gases in $d$ dimensions with $\mathsf{s}\in(\mathsf{d}-2,\mathsf{d})$, tackling the nonlocal fractional interaction through a fractional obstacle problem and a Caffarelli–Silvestre extension. It introduces a transport method tailored to the Riesz setting, couples transport with a screening-based energy bootstrap, and proves local laws down to microscopic scales, alongside a CLT for small powers via an almost-additivity property of the free energy. Central to the approach are sharp commutator estimates for the nonlocal energy, precise control of the transport tails, and a novel analysis of the fractional obstacle problem by Ros-Oton. The results imply convergence of the suitably scaled fluctuations to a fractional Gaussian field in the mesoscopic regime and establish a robust probabilistic description of the system at the smallest relevant scales, with implications for the understanding of nonlocal interacting particle systems and their continuum limits.
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 Leblé 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.