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We study a system of N particles with logarithmic, Coulomb or Riesz pairwiseinteractions, confined by an external potential. We examine a microscopicquantity, the tagged empirical field, for which we prove a large deviationprinciple at speed N. The rate function is the sum of an entropy term, thespecific relative entropy, and an energy term, the renormalized energyintroduced in previous works, coupled by the temperature. We deduce avariational property of the sine-beta processes which arise in random matrixtheory. We also give a next-to-leading order expansion of the free energy ofthe system, proving the existence of the thermodynamic limit.

Large Deviation Principle for Empirical Fields of Log and Riesz Gases