A Note on Truncated Long-Range Percolation with Heavy Tails on Oriented Graphs |
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Authors: | C Alves M R Hilário B N B De Lima D Valesin |
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Institution: | 1.Departamento de Estatística, IMECC,Universidade Estadual de Campinas,Campinas,Brazil;2.Departamento de Matemática,Universidade Federal de Minas Gerais,Belo Horizonte,Brazil;3.Johann Bernoulli Instituut,Rijksuniversiteit Groningen,Groningen,The Netherlands |
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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 x, y 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 \({\mathbb {Z}}^d\). |
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