Calculus of Variations and Geometric Measure Theory

E. Mainini - P. Piovano - B. Schmidt - U. Stefanelli

$N^{3/4}$ law in the cubic lattice

created by mainini on 05 Jul 2018
modified on 24 Sep 2019


Published Paper

Inserted: 5 jul 2018
Last Updated: 24 sep 2019

Journal: J. Stat. Phys.
Volume: 176
Number: 6
Pages: 1480-1499
Year: 2019

ArXiv: 1807.00811 PDF


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 $M_n$ of the edge perimeter are shown to deviate from a corresponding cubic Wulff configuration with respect to their symmetric difference by at most $O(n^{3/4})$ 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.