Homological connectivity of random hypergraphs

TL;DR

This paper investigates the homological connectivity of random hypergraph-derived simplicial complexes, establishing a sharp threshold for when they become homologically connected, and linking this to the disappearance of minimal obstructions.

Contribution

It proves a sharp threshold for homological connectivity in random hypergraph complexes and relates it to the disappearance of minimal obstructions, despite the non-monotone nature of the property.

Findings

Identifies the threshold for homological connectivity in random hypergraph complexes.

Shows the threshold is sharp despite the non-monotonicity of the property.

Establishes a hitting time result linking connectivity to minimal obstruction disappearance.

Abstract

We consider simplicial complexes that are generated from the binomial random 3-uniform hypergraph by taking the downward-closure. We determine when this simplicial complex is homologically connected, meaning that its zero-th and first homology groups with coefficients in $\mathbb{F}_2$ vanish. Although this is not intrinsically a monotone property, we show that it nevertheless has a single sharp threshold, and indeed prove a hitting time result relating the connectedness to the disappearance of the last minimal obstruction.