Duke Mathematical Journal

The Gaussian core model in high dimensions

Henry Cohn and Matthew de Courcy-Ireland

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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 teαt2 with 0<α<4π/e, we show that no point configuration in Rn 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 Rn for the minimal energy under inverse power laws t1/tn+s with s>0, and these bounds are sharp to within a constant factor as n with s fixed.

Article information

Source
Duke Math. J., Volume 167, Number 13 (2018), 2417-2455.

Dates
Received: 12 January 2017
Revised: 21 February 2018
First available in Project Euclid: 14 August 2018

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1534233620

Digital Object Identifier
doi:10.1215/00127094-2018-0018

Mathematical Reviews number (MathSciNet)
MR3855354

Zentralblatt MATH identifier
06970972

Subjects
Primary: 52A40: Inequalities and extremum problems
Secondary: 31C20: Discrete potential theory and numerical methods 52C17: Packing and covering in $n$ dimensions [See also 05B40, 11H31] 82B05: Classical equilibrium statistical mechanics (general)

Keywords
Gaussian core model random lattice linear programming bound

Citation

Cohn, Henry; de Courcy-Ireland, Matthew. The Gaussian core model in high dimensions. Duke Math. J. 167 (2018), no. 13, 2417--2455. doi:10.1215/00127094-2018-0018. https://projecteuclid.org/euclid.dmj/1534233620


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