Harmonicity and invariance on slices of the Boolean cube

Abstract

In a recent work with Kindler and Wimmer we proved an invariance principle for the slice for low-influence, low-degree harmonic multilinear polynomials (a polynomial in \(x_1,\ldots ,x_n\) is harmonic if it is annihilated by \(\sum _{i=1}^n \frac{\partial }{\partial x_i}\)). Here we provide an alternative proof for general low-degree harmonic multilinear polynomials, with no constraints on the influences. We show that any real-valued harmonic multilinear polynomial on the slice whose degree is \(o(\sqrt{n})\) has approximately the same distribution under the slice and cube measures. Our proof is based on ideas and results from the representation theory of \(S_n\), along with a novel decomposition of random increasing paths in the cube in terms of martingales and reverse martingales. While such decompositions have been used in the past for stationary reversible Markov chains, our decomposition is applied in a non-stationary non-reversible setup. We also provide simple proofs for some known and some new properties of harmonic functions which are crucial for the proof. Finally, we provide independent simple proofs for the known facts that (1) one cannot distinguish between the slice and the cube based on functions of o(n) coordinates and (2) Boolean symmetric functions on the cube cannot be approximated under the uniform measure by functions whose sum of influences is \(o(\sqrt{n})\).

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Notes

  1. 1.

    This somewhat unfortunate terminology is borrowed from Bergeron [4, Section 8.4], in which an \(S_n\)-harmonic polynomial is one which is annihilated by \(\sum _{i=1}^n \frac{\partial ^k}{\partial x_i^k}\) for all k. For multilinear polynomials, both definitions coincide.

  2. 2.

    The Lévy distance between two real random variables XY is the infimum value of \(\epsilon \) such that for all \(t \in \mathbb {R}\) it holds that \(\Pr [X \le t - \epsilon ] - \epsilon \le \Pr [Y \le t] \le \Pr [X \le t + \epsilon ] + \epsilon \).

  3. 3.

    The informal term “\(L_2\) theory” refers to studying functions via \(L_2\) norms and inner products. For example, Fourier analysis is an \(L_2\) theory, since its central concept is the Fourier basis, which is an orthonormal basis with respect to a given inner product. In contrast, total variation distance, which is inherently related to coupling, is an \(L_1\) concept.

  4. 4.

    Formally speaking, this step requires us to work with \(\sqrt{I_k}\). However, the ideal \(I_k\) is radical, see for example [34, Lemma 6.1].

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Acknowledgements

Both authors would like to thank the referees for their extensive and helpful comments. Yuval Filmus would like to mention that this material is based upon work supported by the National Science Foundation under Agreement No. DMS-1128155. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors, and do not necessarily reflect the views of the National Science Foundation. Part of the work was done while at the Institute for Advanced Study, Princeton, NJ. The research was also funded by ISF Grant 1337/16. The author is a Taub Fellow, and supported by the Taub Foundations. Elchanan Mossel would like to acknowledge the support of the following Grants: NSF Grants DMS 1106999 and CCF 1320105, DOD ONR Grant N00014-14-1-0823, and Grant 328025 from the Simons Foundation.

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Filmus, Y., Mossel, E. Harmonicity and invariance on slices of the Boolean cube. Probab. Theory Relat. Fields 175, 721–782 (2019). https://doi.org/10.1007/s00440-019-00900-w

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Keywords

  • Analysis of Boolean functions
  • Invariance principle
  • Johnson scheme
  • Slice

Mathematics Subject Classification

  • 60F17
  • 60G99
  • 05E30