We are interested in systems of points with Coulomb, logarithmic or Riesz interactions (i.e. inverse powers of the distance). They arise in various settings: an instance is the study of Fekete points in approximation theory, another is the classical log gas or Coulomb gas which in some cases happens to be a random matrix ensemble, another is vortices in superconductors, superfluids or Bose-Einstein condensates, where one observes the emergence of densely packed point vortices forming perfect triangular lattice patterns named Abrikosov lattices.

After reviewing the motivations, we will take a point of view based on the detailed expansion of the interaction energy to describe the behavior of the systems. In particular a Central Limit Theorem for fluctuations and a Large Deviations Principle for the microscopic point processes are given. This allows to observe the effect of the temperature as it gets very large or very small, and to connect with crystallization questions. The main results are joint with Thomas Leblé and also based on previous works with Etienne Sandier, Nicolas Rougerie and Mircea Petrache.

Presentation by: Sylvia Serfaty. Université Pierre et Marie Curie Paris 6, France.