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Journal of Statistical Physics

, Volume 169, Issue 5, pp 972–980 | Cite as

A Note on Truncated Long-Range Percolation with Heavy Tails on Oriented Graphs

  • C. Alves
  • M. R. Hilário
  • B. N. B. De LimaEmail author
  • D. Valesin
Article

Abstract

We consider oriented long-range percolation on a graph with vertex set \({\mathbb {Z}}^d \times {\mathbb {Z}}_+\) and directed edges of the form \(\langle (x,t), (x+y,t+1)\rangle \), for xy in \({\mathbb {Z}}^d\) and \(t \in {\mathbb {Z}}_+\). Any edge of this form is open with probability \(p_y\), independently for all edges. Under the assumption that the values \(p_y\) do not vanish at infinity, we show that there is percolation even if all edges of length more than k are deleted, for k large enough. We also state the analogous result for a long-range contact process on \({\mathbb {Z}}^d\).

Keywords

Contact processes Oriented percolation Long-range percolation Truncation 

Mathematics Subject Classification

60K35 82B43 

Notes

Acknowledgements

The authors would like to thank Daniel Ungaretti and Rangel Baldasso for helpful discussions. The research of B.N.B.L. was supported in part by CNPq Grant 309468/2014-0 and FAPEMIG (Programa Pesquisador Mineiro). C.A. was supported by FAPESP, Grant 2013/24928-2, and is thankful for the hospitality of the UFMG Mathematics Department. The research of M.H. was partially supported by CNPq Grant 406659/2016-1.

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Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  • C. Alves
    • 1
  • M. R. Hilário
    • 2
  • B. N. B. De Lima
    • 2
    Email author
  • D. Valesin
    • 3
  1. 1.Departamento de Estatística, IMECCUniversidade Estadual de CampinasCampinasBrazil
  2. 2.Departamento de MatemáticaUniversidade Federal de Minas GeraisBelo HorizonteBrazil
  3. 3.Johann Bernoulli InstituutRijksuniversiteit GroningenGroningenThe Netherlands

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