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

C. Alves; M. R. Hilário; B. N.B. De Lima; D. Valesin

doi:10.1007/s10955-017-1899-3

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

C. Alves, M. R. Hilário, B. N.B. De Lima, D. Valesin

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

Abstract

We consider oriented long-range percolation on a graph with vertex set Z^d× Z₊ and directed edges of the form ⟨ (x, t) , (x+ y, t+ 1) ⟩ , for x, y in Z^d and t∈ 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 Z^d.

Original language	English
Pages (from-to)	972-980
Number of pages	9
Journal	Journal of Statistical Physics
Volume	169
Issue number	5
DOIs	https://doi.org/10.1007/s10955-017-1899-3
State	Published - Dec 1 2017
Externally published	Yes

Funding

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.

Funders	Funder number
Fundação de Amparo à Pesquisa do Estado de São Paulo	2013/24928-2, 406659/2016-1
Conselho Nacional de Desenvolvimento Científico e Tecnológico	309468/2014-0
Fundação de Amparo à Pesquisa do Estado de Minas Gerais

Keywords

Contact processes
Long-range percolation
Oriented percolation
Truncation

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10.1007/s10955-017-1899-3

Cite this

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abstract = "We consider oriented long-range percolation on a graph with vertex set Zd× Z+ and directed edges of the form ⟨ (x, t) , (x+ y, t+ 1) ⟩ , for x, y in Zd and t∈ Z+. Any edge of this form is open with probability py, independently for all edges. Under the assumption that the values py 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 Zd.",

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AB - We consider oriented long-range percolation on a graph with vertex set Zd× Z+ and directed edges of the form ⟨ (x, t) , (x+ y, t+ 1) ⟩ , for x, y in Zd and t∈ Z+. Any edge of this form is open with probability py, independently for all edges. Under the assumption that the values py 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 Zd.

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

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