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

TL;DR

This paper studies the edge-isoperimetric problem on the cubic lattice, showing that minimizers closely approximate the Wulff shape with deviations of order n^{3/4}, extending known results to three dimensions.

Contribution

It establishes the optimal O(n^{3/4}) deviation bound for minimizers from the Wulff shape in the cubic lattice, extending 2D lattice results to 3D.

Findings

Deviations from the Wulff shape are at most O(n^{3/4}) elements.

The exponent 3/4 is proven to be optimal.

Results extend previous 2D lattice findings to 3D cubic lattice.

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.