Abstract
We consider different problems within the general theme of long-range percolation on oriented graphs. Our aim is to settle the so-called truncation question, described as follows. We are given probabilities that certain long-range oriented bonds are open; assuming that the sum of these probabilities is infinite, we ask if the probability of percolation is positive when we truncate the graph, disallowing bonds of range above a possibly large but finite threshold. We give some conditions in which the answer is affirmative. We also translate some of our results on oriented percolation to the context of a long-range contact process.
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Acknowledgments
This work was done during B.N.B.L.’s sabbatical stay at IMPA; he would like to thank Rijksuniversiteit Groningen and IMPA for their hospitality. The research of B.N.B.L. was supported in part by CNPq and FAPEMIG (Programa Pesquisador Mineiro).
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van Enter, A.C.D., de Lima, B.N.B. & Valesin, D. Truncated Long-Range Percolation on Oriented Graphs. J Stat Phys 164, 166–173 (2016). https://doi.org/10.1007/s10955-016-1538-4
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DOI: https://doi.org/10.1007/s10955-016-1538-4