Sharp vanishing thresholds for cohomology of random flag complexes

TL;DR

This paper identifies sharp thresholds for the vanishing of cohomology groups and the emergence of property (T) in random flag complexes, revealing phase transitions in their topological and algebraic properties.

Contribution

It precisely determines the sharp vanishing thresholds for cohomology and property (T) in random flag complexes, advancing understanding of their phase transitions.

Findings

Sharp vanishing thresholds for cohomology groups

Threshold for property (T) in fundamental groups

Existence of regimes with rational homotopy equivalence to spheres

Abstract

For every $k \ge 1$, the $k$th cohomology group $H^k(X, \Q)$ of the random flag complex $X \sim X(n,p)$ passes through two phase transitions: one where it appears, and one where it vanishes. We describe the vanishing threshold and show that it is sharp. Using the same spectral methods, we also find a sharp threshold for the fundamental group $\pi_1(X)$ to have Kazhdan's property (T). Combining with earlier results, we obtain as a corollary that for every $k \ge 3$ there is a regime in which the random flag complex is rationally homotopy equivalent to a bouquet of $k$-dimensional spheres.