Abstract
We consider a dependent percolation model on the square lattice . 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 by de Lima (Ph.D.Thesis, Informes de Matemática. IMPA, Série 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 (Comm. Math. Phys. 254 (2005) 1–22).
Funding Statement
During the initial period, B.N.B.L. was partially supported by CNPq grant 301844/2008-9, and M.E.V. was partially supported by CNPq grant 302796/2002-9. During the current revision process, B.N.B.L. is partially supported by CNPq grant 305811/2018-5 and FAPEMIG (Programa Pesquisador Mineiro) and M.E.V. is partially supported by CNPq grant 310734/2021-5 and Faperj grant E-26/202.636/2019.
Acknowledgments
The research that brought to this paper was initiated during the preparation of the Ph.D. thesis of B.N.B.L. It continued over several years that included many inspiring discussions with Harry Kesten (“in memorian”), to whom we are deeply indebted.
Citation
Bernardo N. B. de Lima. Vladas Sidoravicius. Maria Eulália Vares. "Dependent percolation on ." Braz. J. Probab. Stat. 37 (2) 431 - 454, June 2023. https://doi.org/10.1214/23-BJPS575
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