Overcrowding and Separation Estimates for the Coulomb Gas

Abstract

We prove several results for the Coulomb gas in any dimension $d \geq 2$ that follow from isotropic averaging, a transport method based on Newton's theorem. First, we prove a high-density Jancovici-Lebowitz-Manificat law, extending the microscopic density bounds of Armstrong and Serfaty and establishing strictly sub-Gaussian tails for charge excess in dimension $2$. The existence of microscopic limiting point processes is proved at the edge of the droplet. Next, we prove optimal upper bounds on the $k$-point correlation function for merging points, including a Wegner estimate for the Coulomb gas for $k=1$. We prove the tightness of the properly rescaled $k$th minimal particle gap, identifying the correct order in $d=2$ and a three term expansion in $d \geq 3$, as well as upper and lower tail estimates. In particular, we extend the two-dimensional "perfect-freezing regime" identified by Ameur and Romero to higher dimensions. Finally, we give positive charge discrepancy bounds which are state of the art near the droplet boundary and prove incompressibility of Laughlin states in the fractional quantum Hall effect, starting at large microscopic scales. Using rigidity for fluctuations of smooth linear statistics, we show how to upgrade positive discrepancy bounds to estimates on the absolute discrepancy in certain regions.