  We study combinatorial connectivity for two models of random geometric
complexes. These two models - \v{C}ech and Vietoris-Rips complexes - are built
on a homogeneous Poisson point process of intensity $n$ on a $d$-dimensional
torus using balls of radius $r_n$. In the former, the $k$-simplices/faces are
formed by subsets of $(k+1)$ Poisson points such that the balls of radius $r_n$
centred at these points have a mutual interesection and in the latter, we
require only a pairwise intersection of the balls. Given a (simplicial) complex
(i.e., a collection of $k$-simplices for all $k \geq 1$), we can connect
$k$-simplices via $(k+1)$-simplices (`up-connectivity') or via
$(k-1)$-simplices (`down-connectivity). Our interest is to understand these two
combinatorial notions of connectivity for the random \v{C}ech and Vietoris-Rips
complexes asymptically as $n \to \infty$. In particular, we analyse in detail
the threshold radius for vanishing of isolated $k$-faces for up and down
connectivity of both types of random geometric complexes. Though it is expected
that the threshold radius $r_n = \Theta((\frac{\log n}{n})^{1/d})$ in coarse
scale, our results give tighter bounds on the constants in the logarithmic
scale as well as shed light on the possible second-order correction factors.
Further, they also reveal interesting differences between the phase transition
in the \v{C}ech and Vietoris-Rips cases. The analysis is interesting due to the
non-monotonicity of the number of isolated $k$-faces (as a function of the
radius) and leads one to consider `monotonic' vanishing of isolated $k$-faces.
The latter coincides with the vanishing threshold mentioned above at a coarse
scale (i.e., $\log n$ scale) but differs in the $\log \log n$ scale for the
\v{C}ech complex with $k = 1$ in the up-connected case.
