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Line-of-sight percolation
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
2009 (English)In: Combinatorics, probability & computing, ISSN 0963-5483, E-ISSN 1469-2163, Vol. 18, no 1-2, 83-106 p.Article in journal (Refereed) Published
Abstract [en]

Given omega >= 1, let Z((omega))(2) be the graph with vertex Set Z(2) in which two vertices are joined if they agree in one coordinate and differ by at most omega in the other. (Thus Z((1))(2) is precisely Z(2).) Let p(c)(omega) be the critical probability for site percolation on Z((omega))(2) Extending recent results of Frieze, Kleinberg, Ravi and Debany, we show that lim(omega ->infinity) omega p(c)(omega) = log(3/2). We also prove analogues of this result for the n-by-n grid and in higher dimensions, the latter involving interesting connections to Gilbert's continuum percolation model. To prove our results, we explore the component of the origin in a certain non-standard way, and show that this exploration is well approximated by a certain branching random walk.

Place, publisher, year, edition, pages
2009. Vol. 18, no 1-2, 83-106 p.
National Category
Mathematics
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:uu:diva-114321DOI: 10.1017/S0963548308009310ISI: 000265047400005OAI: oai:DiVA.org:uu-114321DiVA: diva2:293734
Available from: 2010-02-12 Created: 2010-02-12 Last updated: 2011-03-11Bibliographically approved

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Janson, Svante

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