  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.
