Mixing in high-dimensional expanders

Abstract: 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. In this paper we remove the assumption of a complete skeleton, showing that concentration of the Laplace spectra in all dimensions implies combinatorial expansion in any complex. As applications we show that spectral concentration implies Gromov's geometric overlap property, and can be used to bound the chromatic number of a complex.