Distribution of components in the \(k\)-nearest neighbour random geometric graph for \(k\) below the connectivity threshold. (English) Zbl 1288.60128

Consider a Poisson point process of intensity 1 in the plane. A random geometric graph \(G\) is defined on the set \(V\) of points of the process inside a square of area \(n\) by joining each point in \(V\) to its \(k\)-nearest neighbours in \(V\). The distribution of small connected components of \(G\) is studied for \(k=k(n)\) below the connectivity threshold. It is also shown that such components are in a specified sense not close together.


60K35 Interacting random processes; statistical mechanics type models; percolation theory
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
05C80 Random graphs (graph-theoretic aspects)
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