Vanishing of cohomology groups of random simplicial complexes

TL;DR

This paper investigates the threshold phenomena for the vanishing of cohomology groups in random simplicial complexes generated from hypergraphs, establishing sharp thresholds, hitting times, and asymptotic distributions for these topological invariants.

Contribution

It identifies sharp thresholds for cohomology vanishing in higher-dimensional random complexes and relates these to the disappearance of minimal obstructions, extending previous results to higher dimensions.

Findings

Sharp threshold for cohomology vanishing established

Hitting time results connect cohomology disappearance to minimal obstructions

Asymptotic distribution of cohomology group dimensions analyzed

Abstract

We consider $k$-dimensional random simplicial complexes that are generated from the binomial random $(k+1)$-uniform hypergraph by taking the downward-closure, where $k\geq 2$. For each $1\leq j \leq k-1$, we determine when all cohomology groups with coefficients in $\mathbb{F}_2$ from dimension one up to $j$ vanish and the zero-th cohomology group is isomorphic to $\mathbb{F}_2$. This property is not deterministically monotone for this model of random complexes, but nevertheless we show that it has a single sharp threshold. Moreover we prove a hitting time result, relating the vanishing of these cohomology groups to the disappearance of the last minimal obstruction. We also study the asymptotic distribution of the dimension of the $j$-th cohomology group inside the critical window. As a corollary, we deduce a hitting time result for a different model of random simplicial complexes introduced in [Linial and Meshulam, Combinatorica, 2006], a result which was previously only known for dimension two [Kahle and Pittel, Random Structures Algorithms, 2016].