Mircea Petrache (Bonn)
Sharp asymptotics and equidistribution for Coulomb and Riesz gases
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
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
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.