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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)

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)

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