  We consider a random geometric graph with vertices sampled from a probability
measure supported on $\mathbb R^d$, and study its connectivity. We show the
graph is typically disconnected, unless the sampling density has
superexponential decay. In the later setting, we identify an asymptotic
threshold value for the radius parameter of the graph such that, for radius
values beyond the threshold, some concentration properties hold for the sampled
points of the graph, while the graph is disconnected for radius values below
the same threshold. Properties of point processes are well-known to be closely
related to the analysis of geometric learning problems, such as spectral
clustering. This work can be seen as a first step towards understanding the
consistency of spectral clustering when the probability measure has unbounded
support. In particular, we narrow down the setting under which spectral
clustering algorithms on $\mathbb R^d$ may be expected to achieve consistency,
to a sufficiently fast decay of the sampling density (superexponential) and a
sufficiently slowly decaying radius parameter value as a function of $n$, the
number of sampled points.
