Next order energy asymptotics for Riesz potentials on flat tori

Abstract

Let $\Lambda$ be a lattice in ${\mathbf{R}}^d$ 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 $-\frac{2}{d}N\log N+\left(C_{\log ,d}-2{\zeta }_{\Lambda }^{\prime }(0)\right)N$ where we show that the constant $C_{s,d}$ is independent of the lattice $\Lambda$.

References in corpus (2)

• Next Order Asymptotics and Renormalized Energy for Riesz Interactions
• Periodic Discrete Energy for Long-Range Potentials

Cited by in corpus (1)

• Dimension reduction techniques for the minimization of theta functions on lattices