Sharp vanishing thresholds for cohomology of random flag complexes [article]

Matthew Kahle
2013 arXiv   pre-print
For every k > 1, the kth cohomology group H^k(X, ) 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 π_1(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
more » ... rationally homotopy equivalent to a bouquet of k-dimensional spheres.
arXiv:1207.0149v3 fatcat:gz2tk43b25gmvmb2kmyzwy2liu