$N^{3/4}$ 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 $\mathrm{O}(n^{3/4})$ 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.

Cited by in corpus (4)

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• Emergence of rigid Polycrystals from atomistic Systems with Heitmann-Radin sticky disk energy
• Maximal fluctuations around the Wulff shape for edge-isoperimetric sets in ${\mathbb{Z}}^d$: a sharp scaling law
• Emergence of Wulff-Crystals from atomistic systems on the FCC and HCP lattices