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We prove several results for the Coulomb gas in any dimension $d \geq 2$ thatfollow from isotropic averaging, a transport method based on Newton's theorem.First, we prove a high-density Jancovici-Lebowitz-Manificat law, extending themicroscopic density bounds of Armstrong and Serfaty and establishing strictlysub-Gaussian tails for charge excess in dimension $2$. The existence ofmicroscopic limiting point processes is proved at the edge of the droplet.Next, we prove optimal upper bounds on the $k$-point correlation function formerging points, including a Wegner estimate for the Coulomb gas for $k=1$. Weprove the tightness of the properly rescaled $k$th minimal particle gap,identifying the correct order in $d=2$ and a three term expansion in $d \geq3$, as well as upper and lower tail estimates. In particular, we extend thetwo-dimensional "perfect-freezing regime" identified by Ameur and Romero tohigher dimensions. Finally, we give positive charge discrepancy bounds whichare state of the art near the droplet boundary and prove incompressibility ofLaughlin states in the fractional quantum Hall effect, starting at largemicroscopic scales. Using rigidity for fluctuations of smooth linearstatistics, we show how to upgrade positive discrepancy bounds to estimates onthe absolute discrepancy in certain regions.

Overcrowding and Separation Estimates for the Coulomb Gas