Gaussian energy minimization in high dimensions

We discuss an energy defined for configurations in Euclidean space and given by a Gaussian interaction between each pair of points. Many other potential functions can be synthesized, by the Laplace transform, as a superposition of Gaussians. The goal is to minimize energy over all configurations with a given number of points per unit volume. Lower bounds for energy can be obtained by the linear programming method whenever one can produce an auxiliary function with certain properties. In the special dimensions 8 and 24, Cohn-Kumar-Miller-Radchenko-Viazovska find the optimal such function and use it to solve the minimization problem exactly. In arbitrary dimension, Cohn and I show how to make a suboptimal choice that leads to a surprisingly good bound. In the easy case where the Gaussian is not too steep, and in the limit of high dimension, this bound asymptotically matches the energy that can be achieved by choosing a random lattice. For a more rapidly decaying Gaussian, the problem is closely related to sphere packing, and there is a wider gap between our upper and lower bounds.