Abstract:

Systems of particles with Coulomb and logarithmic interactions
arise in various settings: an instance is the classical
Coulomb gas which in some cases happens to be a random matrix
ensemble, another is vortices in the Ginzburg-Landau model
of superconductivity, where one observes in certain regimes
the emergence of densely packed point vortices forming perfect
triangular lattice patterns, named Abrikosov lattices in
physics, a third is the study of Fekete points which arise
in approximation theory. I will describe tools to study
such systems and derive a next order (beyond mean field
limit) "renormalized energy" that governs microscopic
patterns of points. I will present the derivation of this
limiting problem, and discuss the question of its minimization
and its link with the Abrikosov lattice and crystallization
questions. I will also discuss generalizations to Riesz
interaction energies, and the statistical mechanics of such
systems.

This is based on joint works with Etienne Sandier, Nicolas
Rougerie, Simona Rota Nodari, Mircea Petrache, and Thomas
Leblé.