Continuum limits of discrete isoperimetric problems and Wulff shapes in lattices and quasicrystal tilings

TL;DR

This paper establishes the convergence of discrete interaction energies on lattices and quasicrystal tilings to a continuous crystalline perimeter, revealing possible Wulff shapes and homogenized perimeters in the continuum limit.

Contribution

It extends discrete-to-continuum convergence results to include quasiperiodic tilings and their associated Wulff shapes using multigrid constructions.

Findings

Discrete energies converge to a crystalline perimeter in the continuum limit.

Wulff shapes can be characterized for various lattice and tiling configurations.

Homogenized perimeters are derived for quasiperiodic tilings via multigrid methods.

Abstract

We prove discrete-to-continuum convergence of interaction energies defined on lattices in the Euclidean space (with interactions beyond nearest neighbours) to a crystalline perimeter, and we discuss the possible Wulff shapes obtainable in this way. Exploiting the "multigrid construction" of quasiperiodic tilings (which is an extension of De Bruijn's "pentagrid" construction of Penrose tilings) we adapt the same techniques to also find the macroscopical homogenized perimeter when we microscopically rescale a given quasiperiodic tiling.