Topology of random d-clique complexes

For a simplicial complex $X$, the $d$-clique complex ${\mathrm{\Delta }}_d(X)$ is the simplicial complex having all subsets of vertices whose $(d+1)$-subsets are contained by $X$ as its faces. We prove that if $p=n^{\alpha }$, with $\alpha < \max{\frac{-1}{k-d +1},-\frac{d+1}{\binom{k}{d}}}$or$\alpha > \frac{-1}{\binom{2k+2}{d}}$,thenthe$k$-th reduced homology group of the random $d$-clique complex ${\mathrm{\Delta }}_d(G_d(n,p))$ is asymptotically almost surely vanishing, and if $\frac{-1}{t}<\alpha <\frac{-1}{t+1}$ where $t = (\frac{(d+1)(k+1)}{\binom{(d+1)(k+1)}{d+1}-(k+1)})^{-1}$,thenthe$(kd + d -1)$-streducedhomologygroupof$\Delta_d(G_d(n,p))$ is asymptotically almost surely nonvanishing. This provides a partial answer to a question posed by Eric Babson.