  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.
