Dependent Percolation on $\mathbb{Z}^2$

TL;DR

This paper proves the existence of a phase transition in a dependent percolation model on the two-dimensional lattice, using a multi-scale renormalization approach and extending previous deterministic environment results.

Contribution

It establishes phase transition for dependent percolation on bZ^2 with infinite-range dependence, expanding on prior deterministic environment work.

Findings

Proved phase transition in dependent percolation on bZ^2.

Developed a multi-scale renormalization framework for dependent models.

Extended results to models with infinite-range dependence.

Abstract

We consider a dependent percolation model on the square lattice $\mathbb{Z}^2$. The range of dependence is infinite in vertical and horizontal directions. In this context, we prove the existence of a phase transition. The proof exploits a multi-scale renormalization argument that is defined once the environment configuration is suitably good and, which, together with the main estimate for the induction step, comes from Kesten, Sidoravicius and Vares (To appear in {\em Electronic Journal of Probability}, (2022)). This work was inspired by de Lima (Ph.D.Thesis, \emph{Informes de Matem\'atica. IMPA}, S\'erie C-26/2004) where the simpler case of a deterministic environment was considered. It has various applications, including an alternative proof for the phase transition on the two dimensional random stretched lattice proved by Hoffman ({\em Comm. Math. Phys.} {\bf 254}, 1-22 (2005)).