  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.
