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Sharp vanishing thresholds for cohomology of random flag complexes. (English) Zbl 1294.05195
Summary: For every \(k \geq 1\), the \(k\)-th cohomology group \(H^k(X, \mathbb{Q})\) 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 \(k \geq 3\), there is a regime in which the random flag complex is rationally homotopy equivalent to a bouquet of \(k\)-dimensional spheres.

05E45 Combinatorial aspects of simplicial complexes
05C80 Random graphs (graph-theoretic aspects)
05C10 Planar graphs; geometric and topological aspects of graph theory
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