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arXiv · 1807.00811

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

Edoardo Mainini·Paolo Piovano·Bernd Schmidt·Ulisse Stefanelli
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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 $ {\rm 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.

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Edoardo Mainini, Paolo Piovano, Bernd Schmidt, Ulisse Stefanelli. 2018-07-01. $N^{3/4}$ law in the cubic lattice. https://doi.org/10.1007/s10955-019-02350-z

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