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On the easiest way to connect k points in the Random Interlacements process
Univ Paris 09, CEREMADE, UMR CNRS 7534, F-75775 Paris 16, France..
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
2013 (English)In: Latin American Journal of Probability and Mathematical Statistics, ISSN 1980-0436, E-ISSN 1980-0436, Vol. 10, no 1, p. 505-524Article in journal (Refereed) Published
Abstract [en]

We consider the random interlacements process with intensity u on Z(d), d >= 5 (call it I-u), built from a Poisson point process on the space of doubly infinite nearest neighbor trajectories on Z(d). For k >= 3 we want to determine the minimal number of trajectories from the point process that is needed to link together k points in I-u. Let n(k, d) := [d/2(k -1)] - (k - 2). We prove that almost surely given any k points x(1),..., x(k) is an element of I-u, there is a sequence of n(k, d) trajectories gamma(1),..., gamma(n(k,d)) from the underlying Poisson point process such that the union of their traces U-i =1(n(k,d)) Tr(gamma(i)) is a connected set containing x(1),..., x(k). Moreover we show that this result is sharp, i.e. that a.s. one can find x(1),..., x(k) is an element of I-u that cannot be linked together by n(k, d) - 1 trajectories.

Place, publisher, year, edition, pages
2013. Vol. 10, no 1, p. 505-524
Keywords [en]
Random Interlacement, Connectivity, Percolation, Random Walk
National Category
Algebra and Logic
Identifiers
URN: urn:nbn:se:uu:diva-309907ISI: 000346350100020OAI: oai:DiVA.org:uu-309907DiVA, id: diva2:1056710
Available from: 2016-12-15 Created: 2016-12-08 Last updated: 2017-11-29Bibliographically approved

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CiteExportLink to record
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Cite
Citation style
  • apa
  • ieee
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  • vancouver
  • Other style
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Language
  • de-DE
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  • nn-NB
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Output format
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