One-sided sharp thresholds for homology of random flag complexes

TL;DR

This paper investigates phase transitions in the homology and fundamental group properties of random flag complexes, revealing intermediate regimes with unique topological features.

Contribution

It establishes new bounds on sharp thresholds for homology and fundamental group properties in random flag complexes, highlighting intermediate regimes with distinct topological phases.

Findings

Existence of a probability regime with nonvanishing homology

Bounds on sharp thresholds for the fundamental group to be free

Identification of an intermediate regime with non-free, non-(T) fundamental groups

Abstract

We prove that the random flag complex has a probability regime where the probability of nonvanishing homology is asymptotically bounded away from zero and away from one. Related to this main result we also establish new bounds on a sharp threshold for the fundamental group of a random flag complex to be a free group. In doing so we show that there is an intermediate probability regime in which the random flag complex has fundamental group which is neither free nor has Kazhdan's property (T).