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We study combinatorial connectivity for two models of random geometriccomplexes. These two models - \v{C}ech and Vietoris-Rips complexes - are builton a homogeneous Poisson point process of intensity $n$ on a $d$-dimensionaltorus using balls of radius $r_n$. In the former, the $k$-simplices/faces areformed 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, werequire 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 twocombinatorial notions of connectivity for the random \v{C}ech and Vietoris-Ripscomplexes asymptically as $n \to \infty$. In particular, we analyse in detailthe threshold radius for vanishing of isolated $k$-faces for up and downconnectivity of both types of random geometric complexes. Though it is expectedthat the threshold radius $r_n = \Theta((\frac{\log n}{n})^{1/d})$ in coarsescale, our results give tighter bounds on the constants in the logarithmicscale as well as shed light on the possible second-order correction factors.Further, they also reveal interesting differences between the phase transitionin the \v{C}ech and Vietoris-Rips cases. The analysis is interesting due to thenon-monotonicity of the number of isolated $k$-faces (as a function of theradius) and leads one to consider `monotonic' vanishing of isolated $k$-faces.The latter coincides with the vanishing threshold mentioned above at a coarsescale (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.

Thresholds for vanishing of `Isolated' faces in random \v{C}ech and Vietoris-Rips complexes