Isoperimetric inequality for nonlocal bi-axial discrete perimeter

TL;DR

This paper introduces and solves a novel nonlocal bi-axial discrete isoperimetric problem, extending classical perimeter concepts to include internal and external components, with implications for long-range Ising models.

Contribution

It defines a new nonlocal perimeter for polyominoes, characterizes its minimizers, and links the problem to metastability in bi-axial Ising models.

Findings

Characterization of minimizers for the nonlocal perimeter

Solution of the isoperimetric problem for fixed area

Connection to metastable behavior in long-range Ising models

Abstract

In the present manuscript we address and solve for the first time a nonlocal discrete isoperimetric problem. We consider indeed a generalization of the classical perimeter, what we call a nonlocal bi-axial discrete perimeter, where, not only the external boundary of a polyomino $\mathcal{P}$ contributes to the perimeter, but all internal and external components of $\mathcal{P}$. Furthermore, we find and characterize its minimizers in the class of polyominoes with fixed area $n$. Moreover, we explain how the solution of the nonlocal discrete isoperimetric problem is related to the rigorous study of the metastable behavior of a long-range bi-axial Ising model.