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A Note on Inhomogeneous Percolation on Ladder Graphs

  • Bernardo N. B. de LimaEmail author
  • Humberto C. Sanna
Article

Abstract

Let \({\mathbb {G}}=({\mathbb {V}},{\mathbb {E}})\) be the graph obtained by taking the cartesian product of an infinite and connected graph \(G=(V,E)\) and the set of integers \({\mathbb {Z}}\). We choose a collection \({\mathcal {C}}\) of finite connected subgraphs of G and consider a model of Bernoulli bond percolation on \({\mathbb {G}}\) which assigns probability q of being open to each edge whose projection onto G lies in some subgraph of \({\mathcal {C}}\) and probability p to every other edge. We show that the critical percolation threshold \(p_{c}(q)\) is a continuous function in (0, 1), provided that the graphs in \({\mathcal {C}}\) are “well-spaced” in G and their vertex sets have uniformly bounded cardinality. This generalizes a recent result due to Szabó and Valesin.

Keywords

Anisotropic percolation Phase transition Critical curves 

Mathematics Subject Classification

60K35 82B43 

Notes

Acknowledgements

BNBL is partially supported by CNPq. Both authors would like to thank CAPES for the financial support.

Funding

This study was funded by Conselho Nacional de Desenvolvimento Científico e Tecnológico (Grant no. 305811/2018-5) and Coordenação de Aperfeiçoamento de Pessoal de Nível Superior.

References

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Copyright information

© Sociedade Brasileira de Matemática 2019

Authors and Affiliations

  1. 1.Departamento de MatemáticaUniversidade Federal de Minas GeraisBelo HorizonteBrazil

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