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Isoperimetric inequality for nonlocal bi-axial discrete perimeter

V. Jacquier, W. M. Ruszel, C. Spitoni

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.

Isoperimetric inequality for nonlocal bi-axial discrete perimeter

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 contributes to the perimeter, but all internal and external components of . Furthermore, we find and characterize its minimizers in the class of polyominoes with fixed area . 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.

Paper Structure

This paper contains 25 sections, 11 theorems, 136 equations, 16 figures.

Table of Contents

  1. Introduction
  2. Notation and Main Result
  3. Proof of Theorem \ref{['main_theorem']}
  4. Auxiliary lemmas about $Per_{\lambda}(\mathcal{P})$
  5. Proof of Proposition \ref{['prop:no_rectangle']}
  6. Proof of Proposition \ref{['prop:rectangle_square']}
  7. Proof of Proposition \ref{['prop:rectangle_strip_attached1']}
  8. Case (i).
  9. Cases (ii).
  10. Case (ii.1).
  11. Case (ii.2).
  12. Proofs of the auxiliary lemmas
  13. Metastability of the two dimensional bi-axial long-range Ising model
  14. The metastability problem
  15. The metastability problem for the bi-axial long-range Ising model
  16. ...and 10 more sections

Key Result

Theorem 1

Fix $n\in \mathbb{N}$. There exists a constant independent of $n$, $\lambda_c$, such that for all $\lambda>\lambda_c >1$ and all polyominos $\mathcal{P}\notin \mathscr{M}_n$ with area $a_{\mathcal{P}} = n$ we have that: for any $\mathcal{R} \in \mathscr{M}_n$.

Figures (16)

  • Figure 1: An example of polyomino with area 25 and classical perimeter 24.
  • Figure 2: Examples of $\mathcal{P}\in\mathscr{M}_n$ for different $n$.
  • Figure 3: Starting from the left hand side, a concave polyomino, a convex polyomino and a cross-convex polyomino.
  • Figure 4: Minimizers of $Per_{\lambda}$ in $\mathscr{M}_n$ for $n\in\{1,\dots, 30\}$.
  • Figure 5: A schematic representation of the proof of Theorem \ref{['main_theorem']}. We distinguish three cases according to the value of $n$. We reduce the set of possible minimizers in any of the cases by each shell. The corresponding proposition doing that is indicated in each shell.
  • ...and 11 more figures

Theorems & Definitions (19)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Theorem 1
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • ...and 9 more