Calculus of Variations and Geometric Measure Theory

A. Braides - A. Chambolle

Ising systems, measures on the sphere, and zonoids

created by braidesa on 18 May 2023
modified on 20 Sep 2023


Accepted Paper

Inserted: 18 may 2023
Last Updated: 20 sep 2023

Journal: Tunisian J. Math.
Year: 2023

ArXiv: 2305.11054 PDF


We give an interpretation of a class of discrete-to-continuum results for Ising systems using the theory of zonoids. We define rational zonotopes and rational zonoids, as the families of Wulff shapes of perimeters obtained as discrete-to-continuum limits of finite-range homogeneous Ising systems and of general homogeneous Ising systems, respectively. Thanks to the characterization of zonoids in terms of measures on the sphere, rational zonotopes can be identified as finite sums of Dirac masses, and hence are dense in the class of all zonoids. Moreover, we show that a rational zonoid can be obtained from a coercive Ising system if and only if the corresponding measure satisfies some ‘connectedness’ properties, while it is always a continuum limit of ‘discrete Wulff shapes’ under the only condition that the support of the measure spans the whole space. Finally, we highlight the connection with the homogenization of periodic Ising systems and propose a generalized definition of rational zonotope of order N, which coincides with the definition of rational zonotope if N=1.

Keywords: Homogenization, perimeter functionals, Ising systems, discrete-to-continuum, zonoids