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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.
##### MSC:
 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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