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We study the graph <i>NN(d,k)</i> constructed on a Poisson point process in <i>d</i> dimensions by connecting each point to the <i>k</i> points nearest to it. We show that if <i>k</i> geq 188, <i>NN(2,k)</i> has an infinite connected component and this infinite component has an infinite subset of points with the property that the distance along the edges of… (More)

We study the graph constructed on a Poisson point process in d dimensions by connecting each point to the k points nearest to it. This graph a.s. has an infinite cluster if k > k c (d) where k c (d), known as the critical value, depends only on the dimension d. This paper presents an improved upper bound of 188 on the value of k c (2). We also show that if… (More)

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