Distribution of components in the \(k\)-nearest neighbour random geometric graph for \(k\) below the connectivity threshold
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Publication:388975
DOI10.1214/EJP.V18-2465zbMATH Open1288.60128arXiv1211.5918OpenAlexW2151967317MaRDI QIDQ388975
Publication date: 17 January 2014
Published in: Electronic Journal of Probability (Search for Journal in Brave)
Abstract: Let S_{n,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 Pr(S_{n,k} connected) > n^{-gamma_1} then the probability that S_{n,k} contains a pair of `small' components `close' to each other is o(n^{-c_1}) (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. (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,k} below the connectivity threshold is asymptotically Poisson.
Full work available at URL: https://arxiv.org/abs/1211.5918
Random graphs (graph-theoretic aspects) (05C80) Interacting random processes; statistical mechanics type models; percolation theory (60K35) Point processes (e.g., Poisson, Cox, Hawkes processes) (60G55)
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