# Scaling transition and edge effects for negatively dependent linear random fields on ${\mathbb{Z}}^2$

## TL;DR

This paper characterizes the anisotropic scaling limits and transition phenomena for negatively dependent linear random fields on Z^2, revealing how edge effects influence the structure of these limits as the domain size grows.

## Contribution

It provides a complete description of scaling limits and transitions for negatively dependent linear random fields with specific decay rates, highlighting the role of edge effects.

## Key findings

- Complete description of anisotropic scaling limits.
- Identification of conditions for scaling transition.
- Edge effects significantly influence the structure of limits.

## Abstract

We obtain a complete description of anisotropic scaling limits and the existence of scaling transition for a class of negatively dependent linear random fields on ${\mathbb{Z}}^2$ with moving-average coefficients $a(t,s)$ decaying as $|t|^{-q_1}$ and $|s|^{-q_2}$ in the horizontal and vertical directions, $q_1^{-1} + q_2^{-1} < 1 $. The scaling limits are taken over rectangles whose sides increase as $\lambda $ and $\lambda^\gamma $ when $\lambda \to \infty$, for any $\gamma >0$. We prove that the scaling transition %and the structure of the scaling diagram in this model is closely related to the presence or absence of the edge effects.