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Mixing in High-Dimensional Expanders


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.

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Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
  • URL: /core/journals/combinatorics-probability-and-computing
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