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 → e −αt 2 with 0 < α < 4π/e, we show that no point configuration in R n of density ρ can have energy less than (ρ + o(1))(π/α) n/2 as n → ∞ with α and ρ fixed. This lower bound asymptotically matches the upper bound of ρ(π/α) 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 ρ(π/α) n/2 is no longer asymptotically sharp when α > πe. As a consequence of our results, we obtain bounds in R n for the minimal energy under inverse power laws t → 1/t n+s with s > 0, and these bounds are sharp to within a constant factor as n → ∞ with s fixed.

Hyperuniform states of matter

“…The Gaussian-core model (GCM), in which particles interact with a a purely repulsive Gaussian pair potential, v 2 (r) = exp[−(r/σ) 2 ], was introduced by Stillinger [364] as a simple model to mimic effective pair interactions between the centers of mass of two polymer chains. Since then, the GCM has been investigated by many groups [363,[365][366][367]. Zachary, Stillinger and Torquato [363] studied the liquid states of this model in R d in various approximations and arbitrary dimensions.…”

A Panorama of Positivity. I: Dimension Free

“…Schoenberg's results were used by Musin [102] to compute the kissing number in four dimensions, by an extension of Delsarte's linear-programming method. Moreover, the results also apply to obtain new bounds on spherical codes [103], with further applications to sphere packing [35,36,37,38]. There are also applications to approximating functions and interpolating data on spheres, pseudodifferential equations with radial basis functions, and Gaussian random fields.…”

Asymptotic Linear Programming Lower Bounds for the Energy of Minimizing Riesz and Gauss Configurations

“…The first is for the large N limit of Riesz energy of N -point configurations on a compact d-rectifiable set embedded in R p , while the second is for the Gaussian energy of infinite configurations in R p having a prescribed density. The latter provides an alternative method for obtaining a main result of Cohn and de Courcy-Ireland [7]. * The research of the authors was supported, in part, by National Science Foundation grant DMS-1516400.…”

“…We next consider bounds for the Gaussian energy of infinite point configurations in R d . Our goal is to show that the method used to prove Theorem 1.4 provides an alternative approach to deriving the lower bounds obtained by Cohn and de Courcy-Ireland [7]. We begin with some essential definitions.…”

“…We shall show that universal lower bounds developed in [5] and based on Delsarte-Levenshtein methods can be used to prove the following estimate of Cohn and de Courcey-Ireland. (The results in [7] came to the authors' attention during the preparation of this manuscript and appear in the dissertation of Michaels [22]. )…”