The Gaussian core model in high dimensions

Abstract

We prove lower bounds for energy in the Gaussian core model, in which point particles interact via a Gaussian potential. Under the potential function $t\mapsto e^{-\alpha t^2}$ with $0<\alpha <4\pi /e$, we show that no point configuration in ${\mathbb{R}}^n$ of density $\rho$ can have energy less than $(\rho +o(1\left)\right)(\pi /\alpha {)}^{n/2}$ as $n\to \infty$ with $\alpha$ and $\rho$ fixed. This lower bound asymptotically matches the upper bound of $\rho (\pi /\alpha {)}^{n/2}$ obtained as the expectation in the Siegel mean value theorem, and it is attained by random lattices. The proof is based on the linear programming bound, and it uses an interpolation construction analogous to those used for the Beurling–Selberg extremal problem in analytic number theory. In the other direction, we prove that the upper bound of $\rho (\pi /\alpha {)}^{n/2}$ is no longer asymptotically sharp when $\alpha >\pi e$. As a consequence of our results, we obtain bounds in ${\mathbb{R}}^n$ for the minimal energy under inverse power laws $t\mapsto 1/t^{n+s}$ with $s>0$, and these bounds are sharp to within a constant factor as $n\to \infty$ with $s$ fixed.