Law in the Cubic Lattice

We investigate the Edge-Isoperimetric Problem (EIP) for sets with n elements of the cubic lattice by emphasizing its relation with the emergence of the Wulff shape in the crystallization problem. Minimizers [Formula omitted] of the edge perimeter are shown to deviate from a corresponding cubic Wulff...

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Bibliographic Details
Published in:Journal of statistical physics 2019-09, Vol.176 (6), p.1480
Main Authors: Mainini, Edoardo, Piovano, Paolo, Schmidt, Bernd, Stefanelli, Ulisse
Format: Article
Language:English
Online Access:Get full text
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Summary:We investigate the Edge-Isoperimetric Problem (EIP) for sets with n elements of the cubic lattice by emphasizing its relation with the emergence of the Wulff shape in the crystallization problem. Minimizers [Formula omitted] of the edge perimeter are shown to deviate from a corresponding cubic Wulff configuration with respect to their symmetric difference by at most [Formula omitted] elements. The exponent 3 / 4 is optimal. This extends to the cubic lattice analogous results that have already been established for the triangular, the hexagonal, and the square lattice in two space dimensions.
ISSN:0022-4715
DOI:10.1007/s10955-019-02350-z