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Mixing in high-dimensional expanders. (English) Zbl 1371.05329
Summary: We establish 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 pseudo-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 pseudo-randomness as in graphs. In this paper we remove the assumption of a complete skeleton, showing that simultaneous concentration of the Laplace spectra in all dimensions implies pseudo-randomness in any complex. We discuss various applications and present some open questions.

05E45 Combinatorial aspects of simplicial complexes
55U10 Simplicial sets and complexes in algebraic topology
58A14 Hodge theory in global analysis
58C40 Spectral theory; eigenvalue problems on manifolds
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