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Distribution of components in the k-nearest neighbour random geometric graph for k below the connectivity threshold
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
2013 (English)In: Electronic Journal of Probability, ISSN 1083-6489, E-ISSN 1083-6489, Vol. 18, no paper 83, 1-22 p.Article in journal (Refereed) Published
Abstract [en]

Let S-n,S-k denote the random geometric graph obtained by placing points inside a square of area n according to a Poisson point process of intensity 1 and joining each such point to the k = k (n) points of the process nearest to it. In this paper we show that if P (S-n,S-k connected) > n(-gamma 1) then the probability that S-n,S-k contains a pair of 'small' components 'close' to each other is o(n (c1)) (in a precise sense of 'small' and 'close'), for some absolute constants gamma(1) > 0 and c(1) > 0. This answers a question of Walters [13]. (A similar result was independently obtained by Balister.) As an application of our result, we show that the distribution of the connected components of S-n,S-k below the connectivity threshold is asymptotically Poisson.

Place, publisher, year, edition, pages
2013. Vol. 18, no paper 83, 1-22 p.
Keyword [en]
Random geometric graphs
National Category
Probability Theory and Statistics
Identifiers
URN: urn:nbn:se:umu:diva-80521DOI: 10.1214/EJP.v18-2465ISI: 000324594600001OAI: oai:DiVA.org:umu-80521DiVA: diva2:649992
Available from: 2013-09-19 Created: 2013-09-19 Last updated: 2017-12-06Bibliographically approved

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