Probability and Stochastic Analysis Seminar 

05/02/2025, 16:00 — 17:00 — Online
Vanessa Jacquier, Utrecht University

Discrete Nonlocal Isoperimetric Inequality and Analysis of the Long-Range Bi-Axial Ising Model

We consider a generalization of the classical perimeter, called 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 $\mathcal{P}$.

Formally, the nonlocal perimeter $Per_{\lambda}(\mathcal{P})$ of the polyomino $\mathcal{P}$ with parameter $\lambda>1$ is defined as:

$$ Per_{\lambda}(\mathcal{P}):=\sum_{x \in \mathbb{Z}^2 \cap \mathcal{P}, \, y \in \mathbb{Z}^2 \cap \mathcal{P}^c} \frac{1}{d^{\lambda}(x,y)} $$

where $d^{\lambda}(x,y)$ is the fractional bi-axial function defined by the relation:

$$ \frac{1}{d^{\lambda}(x,y)} := \frac{1}{|x_2-y_2|^\lambda}\textbf{1}_{\{ x_1=y_1, \, x_2 \neq y_2\}} + \frac{1}{|x_1-y_1|^{\lambda}} \textbf{1}_{\{ x_2=y_2, \, x_1 \neq y_1\}} $$

with $x=(x_1,x_2)$, $y=(y_1,y_2)$ and $\mathcal{P}^c=\mathbb{R}^2 \setminus \mathcal{P}$.

We tackle the nonlocal discrete isoperimetric problem analyzing and characterizing the minimizers within the class of polyominoes with a fixed area $n$.

The solution of this isoperimetric problem provides a foundation for rigorously investigating the metastable behavior of the long-range bi-axial Ising model.