Large deviation principle for empirical fields of Log and Riesz gases

We study the Gibbs measure associated to a system of N particles with logarithmic, Coulomb or Riesz pair interactions under a fairly general confining potential, in the limit N tends to infinity. After rescaling we examine a microscopic quantity, the associated empirical point process, for which we prove a large deviation principle whose rate function is the sum of a specific relative entropy weighted by the temperature and of a "renormalized energy" which measures the disorder of a configuration. This indicates that the configurations should cristallize as the temperature vanishes and behave microscopically like Poisson point processes as the temperature tends to infinity. We deduce a variational characterization of the sine-beta and Ginibre point processes which arise in random matrix theory. We also give a next-to-leading order expansion of the free energy of the system, thus proving the existence of a thermodynamic limit.