A maximum principle for the Coulomb gas: microscopic density bounds, confinement estimates, and high temperature limits

We introduce and prove a maximum principle for a natural quantity related to the $k$-point correlation function of the classical one-component Coulomb gas. As an application, we show that the gas is confined to the droplet by a well-known effective potential in dimensions two and higher. We also prove new upper bounds for the particle density in the droplet that apply at any temperature. In particular, we give the first controls on the microscopic point process for high temperature Coulomb gases beyond the mean-field regime, proving that their laws are uniformly tight in the particle number $N$ for any inverse temperatures $\beta_N$. Furthermore, we prove that limit points are homogeneous mixed Poisson point processes if $\beta_N\to 0$.