The homology of random simplicial complexes in the multi-parameter upper model

We study random simplicial complexes in the multi-parameter upper model. In this model simplices of various dimensions are taken randomly and independently, and our random simplicial complex $Y$ is then taken to be the minimal simplicial complex containing this collection of simplices. We study the asymptotic behavior of the homology of $Y$ as the number of vertices goes to $\mathrm{\infty }$. We observe the following phenomenon asymptotically almost surely. The given probabilities with which the simplices are taken determine a range of dimensions $\mathrm{\ell }\le k\le {\mathrm{\ell }}^{\mathrm{\prime }}$ with ${\mathrm{\ell }}^{\mathrm{\prime }}\le 2\mathrm{\ell }+1$, outside of which the homology of $Y$ vanishes. Within this range, the homologies diminish drastically from dimension to dimension. In particular, the homology in the critical dimension $\mathrm{\ell }$ is significantly the largest.