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Mixing in High-Dimensional Expanders

Published online by Cambridge University Press:  17 May 2017

ORI PARZANCHEVSKI
ORI PARZANCHEVSKI*
Affiliation:
School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540, USA (e-mail: parzan@ias.edu)
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Abstract

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.

Keywords

Primary 05E45Secondary 55U1058A1458C40

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Paper
Information
Combinatorics, Probability and Computing , Volume 26 , Issue 5 , September 2017 , pp. 746 - 761
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Copyright © Cambridge University Press 2017 

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