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Ising systems, measures on the sphere, and zonoids

Andrea Braides and Antonin Chambolle

Vol. 6 (2024), No. 2, 299–319

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, identified as finite sums of Dirac masses, 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.

Ising systems, zonoids, perimeter functionals, discrete-to-continuum, Gamma-convergence, homogenization
Mathematical Subject Classification
Primary: 49J45, 49Q10, 52A20, 82B20
Secondary: 52A40
Received: 13 July 2023
Revised: 20 September 2023
Accepted: 11 October 2023
Published: 29 June 2024
Andrea Braides
Scuola Internazionale Superiore di Studi Avanzati
Antonin Chambolle
Université Paris-Dauphine
PSL University
Paris and Inria Paris (Mokaplan)