Sharp vanishing thresholds for cohomology of random flag complexes

By Matthew Kahle. 2012

102 citations recorded by the source corpus.

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.

Subfields: Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/649b988389e48fb85a92e8d8909e1464410b18da

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Citations in the corpus

What are higher-order networks? (2021, background): “In a series of studies, Kahle and collaborators [156, 158, 159] investigate another construction of complexes based on the Erdős-Rényi (ER) model, called random clique complexes.”
Maximally Persistent Cycles in Random Geometric Complexes (2015, methodology): “P.S. was funded by the EU project TOPOSYS (FP7-ICT-318493-STREP). 1 arXiv:1509.04347v1 [math.PR] 14 Sep 2015 2 OMER BOBROWSKI, MATTHEW KAHLE, AND PRIMOZ SKRABA higher dimensional simplexes (see, e.g. [3, 33, 35, 37]). The study of random simplicial complexes focuses mainly on their homology, which i”
Spectral Gaps of Random Graphs and Applications (2012, methodology): “Applying the same techniques as in the random flag complex paper [33], it is shown that for a certain measure and range of parameter, random right-angled Coxeter groups are rational duality groups w.”
Cliques of Neurons Bound into Cavities Provide a Missing Link between Structure and Function (2016, result): “The obvious utility of the directed flag complex in these applications may also encourage theorists to establish results analogous to those established by Kahle concerning Betti numbers of undirected flag complexes of random graphs [6].”
Limit theorems for Betti numbers of random simplicial complexes (2010, methodology): “In fact, the techniques in [1] give that for p in the given regime, βk = β̃k with probability 1− o(n−M ) for any constant M > 0.”
Random Simplicial Complexes (2014, result): “Kahle [9], [10] and by some other authors.”