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Crystallization for Coulomb and Riesz interactions as a consequence of the Cohn-Kumar conjecture


Authors: Mircea Petrache and Sylvia Serfaty
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
Published electronically: March 23, 2020
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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
Email: decostruttivismo@gmail.com

Sylvia Serfaty
Affiliation: Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, New York 10012
Email: serfaty@cims.nyu.edu

DOI: https://doi.org/10.1090/proc/15003
Keywords: Crystallization, Cohn-Kumar conjecture, Coulomb interaction, Riesz interaction, jellium, renormalized energy, Abrikosov lattice, triangular lattice, Wigner crystal
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
Article copyright: © Copyright 2020 American Mathematical Society