Vanishing of cohomology groups of random simplicial complexes
| Autor: | Cooley, Oliver, Del Giudice, Nicola, Kang, Mihyun, Sprüssel, Philipp |
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| Rok vydání: | 2018 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | We consider $k$-dimensional random simplicial complexes that are generated from the binomial random $(k+1)$-uniform hypergraph by taking the downward-closure, where $k\geq 2$. For each $1\leq j \leq k-1$, we determine when all cohomology groups with coefficients in $\mathbb{F}_2$ from dimension one up to $j$ vanish and the zero-th cohomology group is isomorphic to $\mathbb{F}_2$. This property is not deterministically monotone for this model of random complexes, but nevertheless we show that it has a single sharp threshold. Moreover we prove a hitting time result, relating the vanishing of these cohomology groups to the disappearance of the last minimal obstruction. We also study the asymptotic distribution of the dimension of the $j$-th cohomology group inside the critical window. As a corollary, we deduce a hitting time result for a different model of random simplicial complexes introduced in [Linial and Meshulam, Combinatorica, 2006], a result which was previously only known for dimension two [Kahle and Pittel, Random Structures Algorithms, 2016]. Comment: 35 pages |
| Databáze: | arXiv |
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