N 3 / 4 Law in the 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 $O\left(n^{3/4}\right)$ 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.

Keywords: N 3 / 4 law; Cubic lattice; Edge perimeter; Fluctuations; Wulff shape.

Fig. 1

The daisies$D_d$, d increases from left to right

Fig. 2

Configuration $M_n^{\prime }$. A caveat: in favor of illustrative clarity, proportions in this and the following figures do not correspond to the actual ones of a ground state

Fig. 3

Configuration $M_n^{\prime \prime }$

Fig. 4

Configuration $M_n^{\prime \prime \prime }$

Fig. 5

Configuration $M_{n_s}^{\prime }$

Fig. 6

The exponent 3 / 4 is not optimal for $d\ge 5$: configurations $Q_1$ and $Q_2$