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

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.

27 pages, 2 figures