The two-dimensional one-component plasma is hyperuniform

We prove that at all positive temperatures in the bulk of a classical two-dimensional one-component plasma (also called Coulomb or log-gas, or jellium) the variance of the number of particles in large disks grows more slowly than the area. In other words the system is hyperuniform. We obtain a non-sharp but quantitative bound on the number variance's growth rate, which is the first mathematical justification of an old prediction in the physics literature about "suppression of charge fluctuations". We introduce an argument of approximate conditional independence for well-separated sub-systems and a trick using "isotropically averaged localized translations" in order to control the expectation of non-smooth linear statistics.