Abstract:

We consider the statistical mechanics of a classical Coulomb or Riesz gas in general dimension d. This topic appears among others in random matrix models, in the study of Fekete points in constructive approximation, and in Ginzburg-Landau vortex models for superconductors.

In an asymptotics of the minimum energy gasses going beyond the leading order mean-field limit, a "renormalized energy" W appears. This is a Coulombian or Riesz-type "jellium" interaction energy for an infinite set of point charges in the plane placed in a unform neutralizing backgound.

Jointly with Sylvia Serfaty we obtained in a general setting the characterization of the behavior of the system at the microscopic scale, obtaining that as the temperature tends to zero the system "crystallizes" to a minimizer of W.

The minimum of W is expected to be achieved by the "Abrikosov" triangular lattice in 2 dimensions, whereas in high dimension we find links to the open problems in the study of theta and zeta functions, for which the periodicity of minimizers is also not known. With Simona Rota-Nodari we then proved sharp microscale discrepancy bounds for sequences of Coulomb energy minimizers in general dimension, akin to a strong version of hyperuniformity. This gives a sharp quantitative distinction between microscale behavior of minimizers and that of random configurations.