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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Crystallization for Coulomb and Riesz interactions as a consequence of the Cohn-Kumar conjecture
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by Mircea Petrache and Sylvia Serfaty PDF
Proc. Amer. Math. Soc. 148 (2020), 3047-3057 Request permission

Abstract:

The Cohn-Kumar conjecture states that the triangular lattice in dimension 2, the $E_8$ lattice in dimension 8, and the Leech lattice in dimension 24 are universally minimizing in the sense that they minimize the total pair interaction energy of infinite point configurations for all completely monotone functions of the squared distance. This conjecture was recently proved by Cohn-Kumar-Miller-Radchenko-Viazovska in dimensions 8 and 24. We explain in this note how the conjecture implies the minimality of the same lattices for the Coulomb and Riesz renormalized energies as well as jellium and periodic jellium energies, hence settling the question of their minimization in dimensions 8 and 24.
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Additional Information
  • Mircea Petrache
  • Affiliation: Facultad de Matematicas, PUC Chile, Av. Vicuna Mackenna 4860, 6904441, Santiago, Chile
  • MR Author ID: 907727
  • Email: decostruttivismo@gmail.com
  • Sylvia Serfaty
  • Affiliation: Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, New York 10012
  • MR Author ID: 637763
  • Email: serfaty@cims.nyu.edu
  • Received by editor(s): September 6, 2019
  • Published electronically: March 23, 2020
  • Additional Notes: The first author was supported by the Fondecyt Iniciación grant number 11170264 entitled “Sharp asymptotics for large particle systems and topological singularities”.
    The second author was supported by NSF grant DMS-1700278 and by a Simons Investigator grant.
  • Communicated by: Matthew A. Papanikolas
  • © Copyright 2020 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 148 (2020), 3047-3057
  • MSC (2010): Primary 52C35, 52C99, 82C05, 82C22, 11H06, 11H31
  • DOI: https://doi.org/10.1090/proc/15003
  • MathSciNet review: 4099791