Isoperimetric inequality for nonlocal bi-axial discrete perimeter

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.

💡 Research Summary

The paper tackles a novel discrete isoperimetric problem in two dimensions. Instead of the classical perimeter, which counts only the edges separating a polyomino from its complement, the authors introduce a non‑local bi‑axial perimeter (P^{\mathrm{er}}_{\lambda}). For a polyomino (P\subset\mathbb Z^2) and a parameter (\lambda>1) they define
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