# Random Simplicial Complexes, Duality and The Critical Dimension

## TL;DR

This paper explores dual models of random simplicial complexes, analyzing their homological properties and introducing new notions of critical dimension and spread, with results on Betti number behavior.

## Contribution

It introduces a duality between lower and upper models of random simplicial complexes and extends the concept of critical dimension to the upper model.

## Key findings

- Betti numbers in the upper model are characterized by a new critical dimension.
- Upper models are homologically approximated by a wedge of spheres of the critical dimension.
- Duality relates the homological behavior of lower and upper models.

## Abstract

In this paper we discuss two general models of random simplicial complexes which we call the lower and the upper models. We show that these models are dual to each other with respect to combinatorial Alexander duality. The behaviour of the Betti numbers in the lower model is characterised by the notion of critical dimension, which was introduced by A. Costa and M. Farber: random simplicial complexes in the lower model are homologically approximated by a wedge of spheres of dimension equal the critical dimension. In this paper we study the Betti numbers in the upper model and introduce new notions of critical dimension and spread. We prove that (under certain conditions) an upper random simplicial complex is homologically approximated by a wedge of spheres of the critical dimension.