Random Steiner systems and bounded degree coboundary expanders of every dimension

Abstract: We introduce a new model of random $d$-dimensional simplicial complexes, for $d\geq 2$, whose $(d-1)$-cells have bounded degrees. We show that with high probability, complexes sampled according to this model are coboundary expanders. The construction relies on Keevash's recent result on designs [Ke14], and the proof of the expansion uses techniques developed by Evra and Kaufman in [EK15]. This gives a full solution to a question raised in [DK12], which was solved in the two-dimensional case by Lubotzky and Meshulam [LM13].