Sharp deviation inequalities for the 2D Coulomb gas and Quantum hall states, I

We establish sharp deviation inequalities for the linear statistics of the 2D Coulomb gas. These imply sub-Gaussian inequalities, where the variance is given by the Dirichlet norm. The proofs use complex geometry and potential theory on Riemann surfaces and apply more generally to beta-ensembles, which also include integer Quantum Hall states on Riemann surfaces. In a sequel of the paper we give applications to large and moderate deviation principles, local laws at mesoscopic scales, quantitative Bergman kernel asymptotics. In a series of companion papers applications to concentration of measure, Monte-Carlo methods for numerical integration and random matrices are given and relations to Kahler geometry are explored.