Random Steiner systems and bounded degree coboundary expanders of every dimension

TL;DR

This paper introduces a new model for random high-dimensional simplicial complexes with bounded degrees, demonstrating that these complexes are coboundary expanders with high probability, thus advancing understanding in high-dimensional combinatorics.

Contribution

It presents a novel random model for bounded degree simplicial complexes and proves they are coboundary expanders, solving a longstanding open problem in higher dimensions.

Findings

Complexes are coboundary expanders with high probability.

The model relies on Keevash's design results.

The proof adapts techniques from Evra and Kaufman.

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].