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 ( †) Supported by The Fund for Math at the Institute for Advanced Study.

Local Spectral Expansion Approach to High Dimensional Expanders Part II: Mixing and Geometrical Overlapping

“…The error term in that work was of the form |U 0 ||U n ||U 1 |...|U n−1 |. Also, a criterion for mixing for general (non-random) complexes based on the spectra of all high dimensional Laplacians was proven by Parzanchevski [Par17] and the error term in that result was max i |U i |. Our results improve the error terms in both works.…”

Hypergraph expanders from Cayley graphs

“…A general result of this type was proven by Parzanchevski [25], saying that the desired pseudorandomness condition holds under essentially the hypothesis on the spectra of all the adjacency matrices that we proved in Theorem 3.9. However, Parzanchevski's result only applies when |λ2| λ1 < ǫ 0 (r) where ǫ 0 (r) is a small constant depending on the uniformity, while we have only proved that |λ2| λ1 < 1 − c(r)ǫ where ǫ is the expansion parameter of the original Cayley graph and c(r) is a very small constant depending on the uniformity.…”

“…When ǫ is sufficiently close to 1 in terms of r, an argument in [25,26] allows us to deduce that H r,t,S is a geometric expander. The idea is simple.…”

Local Spectral Expansion Approach to High Dimensional Expanders Part I: Descent of Spectral Gaps

“…In [Lub14] two main approaches are suggested: The first is through the F 2 -coboundary expansion of X originated in [Gro10], [LM06] and [MW09] . The second is through studying the spectral gap of the (n − 1)-Laplacian of X (where n is the dimension of X) or the spectral gaps of all 0, .., (n − 1)-Laplacians of X (see [Par17], [PRT16]). One of the difficulties with both approaches are that both the F 2 -coboundary expansion and the spectral gap of the (n − 1)-Laplacian are usually hard to calculate or even bound in examples.…”