Sharp vanishing thresholds for cohomology of random flag complexes

For every k ≥ 1, the k-th cohomology group H(X,Q) of the random flag complex X ∼ 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 π1(X) to have Kazhdan’s property (T). Combining with earlier results, we obtain as a corollary that for every k ≥ 3, there is a regime in which the random flag complex is rationally homotopy equivalent to a bouquet of k-dimensional spheres.