Bounds and Limiting Minimizers for a Family of Interaction Energies

Abstract

We study a two parameter family of energy minimization problems for interaction energies $\mathcal{E}_{α,β}$ with attractive-repulsive potential $W_{α,β}$. We develop a concavity principle, which allows us to provide a lower bound on $\mathcal{E}_{α,β}$ if there exist $β_0<β<β_1$ with minimizers of $\mathcal{E}_{α,β_0}$ and $\mathcal{E}_{α,β_1}$ known. In addition to this, we also derive new conclusions about the limiting behaviour of $\mathcal{E}_{α,β}$ for $β\approx 2.$ Finally, we describe a method to show that, for certain values of $(α,β),$ $\mathcal{E}_{α,β}$ 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 $\mathcal{E}_{α,β}$ for a range of $α$ and $β$, and an analysis of $\inf\mathcal{E}_{α,β}$ as a function on parameter space.