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15 September 2018 The Gaussian core model in high dimensions
Henry Cohn, Matthew de Courcy-Ireland
Duke Math. J. 167(13): 2417-2455 (15 September 2018). DOI: 10.1215/00127094-2018-0018

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.

Citation

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Henry Cohn. Matthew de Courcy-Ireland. "The Gaussian core model in high dimensions." Duke Math. J. 167 (13) 2417 - 2455, 15 September 2018. https://doi.org/10.1215/00127094-2018-0018

Information

Received: 12 January 2017; Revised: 21 February 2018; Published: 15 September 2018
First available in Project Euclid: 14 August 2018

zbMATH: 06970972
MathSciNet: MR3855354
Digital Object Identifier: 10.1215/00127094-2018-0018

Subjects:
Primary: 52A40
Secondary: 31C20, 52C17, 82B05

Rights: Copyright © 2018 Duke University Press

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Vol.167 • No. 13 • 15 September 2018
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