Hypercontractivity on High Dimensional Expanders: a Local-to-Global Approach for Higher Moments

Abstract

Hypercontractivity is one of the most powerful tools in Boolean function analysis. Originally studied over the discrete hypercube, recent years have seen increasing interest in extensions to settings like the p-biased cube, slice, or Grassmannian, where variants of hypercontractivity have found a number of breakthrough applications including the resolution of Khot's 2-2 Games Conjecture (Khot, Minzer, Safra FOCS 2018). In this work, we develop a new theory of hypercontractivity on high dimensional expanders (HDX), an important class of expanding complexes that has recently seen similarly impressive applications in both coding theory and approximate sampling. Our results lead to a new understanding of the structure of Boolean functions on HDX, including a tight analog of the KKL Theorem and a new characterization of non-expanding sets.Unlike previous settings satisfying hypercontractivity, HDX can be asymmetric, sparse, and very far from products, which makes the application of traditional proof techniques challenging. We handle these barriers with the introduction of two new tools of independent interest: a new explicit combinatorial Fourier basis for HDX that behaves well under restriction, and a new local-to-global method for analyzing higher moments. Interestingly, unlike analogous second moment methods that apply equally across all types of expanding complexes, our tools rely inherently on simplicial structure. This suggests a new distinction among high dimensional expanders based upon their behavior beyond the second moment.

Eigenstripping, Spectral Decay, and Edge-Expansion on Posets

“…It remains an open problem whether a k-independent version can be proved for any q-eposet beyond the Grassmann poset itself. We conjecture such a result should indeed hold (albeit under a different notion of pseudorandomness), and may follow from q-analog analysis of recent work proving k-independent bounds for standard expanding hypergraphs [35,36].…”

“…To our knowledge, the same proof cannot be used, for instance, to resolve the related "shortcode expansion hypotheses" beyond degree-2, similar conjectures offered by Barak,Kothari,and Steurer [29] in an effort to push beyond hardness of 2-2 Games. Just as the ℓ 2 -regime analysis of DDFH and BHKL recently lead to a dimension independent bound in the ℓ 8 -regime for standard HDX [35,36], we expect the groundwork laid in this paper will be important for proving generalized dimension independent expansion hypotheses in the ℓ 8 -regime beyond the special case of the Grassmann graphs.…”

“…We note that this does not recover the result used for the proof of the 2-2 Games Conjecture which lies in the ℓ 8 -regime (replacing variance above with maximum) and requires a dimension-independent bound. This issue was recently (and independently) resolved for simplicial complexes in [35] and [36], and we view our work as an important step towards a more general understanding for families like the Grassmann beyond hypergraphs.…”

“…Furthermore, it is well known the dependence on k in this result is necessary [26], even if one is willing to suffer a worse dependence on the pseudorandomness ε. This is different from the case of standard simplicial complexes, where the dependence can be removed in the ℓ 8 -regime [30,35,36]. However, there is a crucial subtlety here.…”

“…While our definition of pseudorandomness is weaker than that of [25] and therefore necessarily depends on the dimension k, we take the above as evidence that the framework of expanding posets may be important for making further progress on the Unique Games Conjecture. In particular, combined with recent works removing this k-dependence on simplicial complexes [35,36], it seems plausible that the framework of expanding posets may lead to a more general understanding of the structure underlying the unique games conjecture.…”