# Random Simplicial Complexes in the Medial Regime

## TL;DR

This paper analyzes the topology of random simplicial complexes in the medial regime, revealing that their Betti numbers are concentrated in narrow dimension ranges and introducing a new Alexander duality-based technique.

## Contribution

It provides a detailed topological analysis of random simplicial complexes in the medial regime and develops a novel method using Alexander duality to relate lower and upper models.

## Key findings

- Betti numbers of complexes are confined to narrow dimension ranges
- Upper complexes are with high probability non-vanishing only in specific Betti dimensions
- Lower complexes are highly connected with dimension scaling as log-log of the number of vertices

## Abstract

We describe topology of random simplicial complexes in the lower and upper models in the medial regime, i.e. under the assumption that the probability parameters $p_\sigma$ approach neither $0$ nor $1$. We show that nontrivial Betti numbers of typical lower and upper random simplicial complexes in the medial regime lie in a narrow range of dimensions. For instance, an upper random simplicial complex $Y$ on $n$ vertices in the medial regime with high probability has non-vanishing Betti numbers $b_{j}(Y)$ only for $k+c <n-j<k+\log_2 k +c'$ where $k=\log_2 \ln n$ and $c, c' $ are constants. A lower random simplicial complex on $n$ vertices in the medial regime is with high probability $(k+a)$-connected and its dimension $d$ satisfies $d\sim k+\log_2 k+ a'$ where $a, \, a'$ are constants. The paper develops a new technique, based on Alexander duality, which relates the lower and upper models.