Bounds and Limiting Minimizers for a Family of Interaction Energies

Abstract

We study a two parameter family of energy minimization problems for interaction energies ${\mathcal{E}}_{\alpha ,\beta }$ with attractive-repulsive potential $W_{\alpha ,\beta }$. We develop a concavity principle, which allows us to provide a lower bound on ${\mathcal{E}}_{\alpha ,\beta }$ if there exist ${\beta }_0<\beta <{\beta }_1$ with minimizers of ${\mathcal{E}}_{\alpha ,{\beta }_0}$ and ${\mathcal{E}}_{\alpha ,{\beta }_1}$ known. In addition to this, we also derive new conclusions about the limiting behaviour of ${\mathcal{E}}_{\alpha ,\beta }$ for $\beta \approx 2.$ Finally, we describe a method to show that, for certain values of $(\alpha ,\beta ),$ ${\mathcal{E}}_{\alpha ,\beta }$ cannot be minimized by the uniform distribution over a top-dimensional regular unit simplex. Our results are made possible by two key factors -- recent progress in identifying minimizers of ${\mathcal{E}}_{\alpha ,\beta }$ for a range of $\alpha$ and $\beta$, and an analysis of $\inf {\mathcal{E}}_{\alpha ,\beta }$ as a function on parameter space.

22 pages, 1 figure