Crystallinity of the homogenized energy density of periodic lattice systems

TL;DR

This paper proves that the homogenized energy density of periodic ferromagnetic Ising systems with finite range interactions is crystalline, with a polytope Wulff shape, and provides a dual formula for easy numerical computation.

Contribution

It establishes the crystalline nature of the homogenized energy density and derives a dual representation enabling efficient numerical analysis.

Findings

Homogenized energy density is crystalline with a polytope Wulff shape.

Finite cell formula allows explicit computation of the energy density.

Numerical experiments optimize surface tension anisotropy.

Abstract

We study the homogenized energy densities of periodic ferromagnetic Ising systems. We prove that, for finite range interactions, the homogenized energy density, identifying the effective limit, is crystalline, i.e. its Wulff crystal is a polytope, for which we can (exponentially) bound the number of vertices. This is achieved by deriving a dual representation of the energy density through a finite cell formula. This formula also allows easy numerical computations: we show a few experiments where we compute periodic patterns which minimize the anisotropy of the surface tension.