  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.
