Published Paper
Inserted: 5 jul 2018
Last Updated: 24 sep 2019
Journal: J. Stat. Phys.
Volume: 176
Number: 6
Pages: 1480-1499
Year: 2019
Doi: doi.org/10.1007/s10955-019-02350-z
Abstract:
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.