Discrete Minimal Energy on Flat Tori and Four-Point Maximal Polarization on S^2

Abstract

Let $Lambda$ be a lattice in $R^d$ with positive co-volume. Among $Lambda$-periodic $N$-point configurations, we consider the minimal renormalized Riesz $s$-energy $mathcal{E}{s,Lambda}(N)$. While the dominant term in the asymptotic expansion of $mathcal{E}{s,Lambda}(N)$ as $N$ goes to infinity in the long range case that $0<s<d$ (or $s=log$) can be obtained from classical potential theory, the next order term(s) require a different approach. Here we derive the form of the next order term or terms, namely for $s>0$ they are of the form $C_{s,d}|Lambda|^{-s/d}N^{1+s/d}$ and $-frac{2}{d}Nlog N+left(C_{log,d}-2zeta'{Lambda}(0) ight)N$ where we show that the constant $C{s,d}$ is independent of the lattice $Lambda$. We also solve the $4$-point maximal polarization problem on $S^2$. We prove that the vertices of a regular tetrahedron on $S^2$ maximize the minimum of discrete potentials on $S^2$ whenever the potential is of the form $sumlimits_{k=1}^{4}f(|x-x_k|^2)$, where $f:[0,4] ightarrow[0,infty]$ is non-increasing and strictly convex with $f(0)=limlimits_{x o 0^+}f(x)$.