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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.

##### MSC:
 05E45 Combinatorial aspects of simplicial complexes 05C80 Random graphs (graph-theoretic aspects) 05C10 Planar graphs; geometric and topological aspects of graph theory
##### Keywords:
random topology; sharp thresholds
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##### References:
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