Dependent percolation on Z2

We consider a dependent percolation model on the square lattice Z2. 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 (Electronic Journal of Probability 27 (2022) 1-49). This paper is inspired 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 (Comm. Math. Phys. 254 (2005) 1-22).