Topology of random d-clique complexes

Abstract

For a simplicial complex $X$, the $d$-clique complex ${\Delta }_d(X)$ 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 $\alpha <\max \{\frac{-1}{k-d+1},-\frac{d+1}{\left(\genfrac{}{}{0ex}{}{k}{d}\right)}\}$ or $\alpha >\frac{-1}{\left(\genfrac{}{}{0ex}{}{2k+2}{d}\right)}$, then the $k$-th reduced homology group of the random $d$-clique complex ${\Delta }_d(G_d(n,p))$ is asymptotically almost surely vanishing, and if $\frac{-1}{t}<\alpha <\frac{-1}{t+1}$ where $t=(\frac{(d+1)(k+1)}{\left(\genfrac{}{}{0ex}{}{(d+1)(k+1)}{d+1}\right)-(k+1)}{)}^{-1}$, then the $(kd+d-1)$-st reduced homology group of ${\Delta }_d(G_d(n,p))$ is asymptotically almost surely nonvanishing. This provides a partial answer to a question posed by Eric Babson.

11 pages

References in corpus (1)

• Fundamental Groups of Random Clique Complexes

Cited by in corpus (1)

• Law of large numbers for Betti numbers of homogeneous and spatially independent random simplicial complexes