Mixing in high-dimensional expanders

Understand

We prove a generalization of the Expander Mixing Lemma for arbitrary (finite) simplicial complexes.

• The original lemma states that concentration of the Laplace spectrum of a graph implies combinatorial expansion (which is also referred to as mixing, or quasi-randomness).
• Recently, an analogue of this Lemma was proved for simplicial complexes of arbitrary dimension, provided that the skeleton of the complex is complete.
• More precisely, it was shown that a concentrated spectrum of the simplicial Hodge Laplacian implies a similar type of expansion as in graphs.