We study the size of connected components of random nearest-neighbor graphs with vertex set the points of a homogeneous Poisson point process in ℝd. 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.
"The size of components in continuum nearest-neighbor graphs." Ann. Probab. 34 (2) 528 - 538, March 2006. https://doi.org/10.1214/009117905000000729