Asymptotics of particles minimizing power-law interactions: results and questions

I will describe a result obtained with Sylvia Serfaty about asymptotic limits of particle systems in $\mathbb R^d$ with pairwise interactions modeled by Riesz kernels $|x|^{-s}$ for $s \in [\max\{0,d-2\},d[$.

Motivations for such choices of $s$ arise in several fields, including Coulomb gases, eigenvalues for some random matrix ensembles, Fekete sets and spherical designs from approximation theory and the physics of seminconductors immersed in strong magnetic fields. I will briefly recall how the first order term in the asymptotic expansion of the equilibrium energy (the mean field limit) can be obtained. Then I will show how to study and control a next order "renormalized energy" that governs microscopic patterns of points and is an energy of infinite configurations of points.

In the second half of the talk I will explain some open problems, namely 1) The long-stanting open questions about crystallization at the microscopic scale. 2) The case of interactions corresponding to the more non-local "fat tailed" interaction energies with $s\in[0,d-2[$. 3) Some GMT open questions in the case of integer $s$ under the constraint that the points belong to a fixed s-rectifiable set.