Geodesic
Sharp vanishing thresholds for cohomology of random flag complexes
Matthew Kahle1
1 Department of Mathematics, The Ohio State University, Columbus, OH 43202
Annals of mathematics, Tome 179 (2014) no. 3, pp. 1085-1107.

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For every $k \ge 1$, 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 $k \ge 3$, there is a regime in which the random flag complex is rationally homotopy equivalent to a bouquet of $k$-dimensional spheres.
DOI : 10.4007/annals.2014.179.3.5

Matthew Kahle 1

1 Department of Mathematics, The Ohio State University, Columbus, OH 43202 
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Matthew Kahle. Sharp vanishing thresholds for cohomology of random flag complexes. Annals of mathematics, Tome 179 (2014) no. 3, pp. 1085-1107. doi : 10.4007/annals.2014.179.3.5. http://geodesic.mathdoc.fr/articles/10.4007/annals.2014.179.3.5/

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