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

TL;DR

This paper investigates the asymptotic homology behavior of random simplicial complexes generated in a multi-parameter upper model, revealing a dimension range where homology persists and highlighting the critical dimension with maximal homology.

Contribution

It introduces a detailed analysis of homology in multi-parameter random simplicial complexes, identifying the dimension range and the critical dimension with maximal homology.

Findings

Homology vanishes outside a specific dimension range.

Within the range, homology diminishes from higher to lower dimensions.

The critical dimension exhibits the largest homology.

Abstract

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 $\infty$. We observe the following phenomenon asymptotically almost surely. The given probabilities with which the simplices are taken determine a range of dimensions $\ell \leq k \leq \ell'$ with $\ell' \leq 2\ell +1$, 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 $\ell$ is significantly the largest.