The size of components in continuum nearest-neighbor graphs

Details The size of components in continuum nearest-neighbor graphs The size of components in continuum nearest-neighbor graphs

2006-05-24

arxiv.org/abs/math/0605640v1

Annals of Probability 2006, Vol. 34, No. 2, 528-538

Mathematics

Probability

Published at http://dx.doi.org/10.1214/009117905000000729 in the Annals of Probability (http://www.imstat.org/aop/) by the Ins

Scientific paper

10.1214/009117905000000729

We study the size of connected components of random nearest-neighbor graphs with vertex set the points of a homogeneous Poisson point process in ${\mathbb{R}}^d$. The connectivity function is shown to decay superexponentially, and we identify the exact exponent. From this we also obtain the decay rate of the maximal number of points of a path through the origin. We define the generation number of a point in a component and establish its asymptotic distribution as the dimension $d$ tends to infinity.

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