## 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 \(o(\sqrt{n})\).

This is a preview of subscription content, log in to check access.

## Notes

- 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.
The Lévy distance between two real random variables

*X*,*Y*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.
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.
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].

## References

- 1.
Ambainis, A., Belovs, A., Regev, O., de Wolf, R.: Efficient quantum algorithms for (gapped) group testing and junta testing. In: Proceedings of the twenty-seventh annual ACM-SIAM symposium on discrete algorithms (SODA ’16), pp. 903–922 (2016)

- 2.
Bannai, E., Ito, T.: Algebraic Combinatorics I: Association Schemes. Mathematics Lecture Notes Series. Benjamin/Cummings, San Francisco (1984)

- 3.
Beckner, W.: Inequalities in Fourier analysis. Ann. Math.

**102**, 159–182 (1975) - 4.
Bergeron, F.: Algebraic Combinatorics and Coinvariant Spaces. CMS Treatises in Mathematics. A K Peters, Boca Raton (2009)

- 5.
Blekherman, G.: Symmetric sums of squares on the hypercube. Manuscript in preparation (2015)

- 6.
Bonami, A.: Étude des coefficients Fourier des fonctions de \(L^p(G)\). Ann. Inst. Fourier

**20**(2), 335–402 (1970) - 7.
Boppana, R.B.: The average sensitivity of bounded-depth circuits. Inf. Process. Lett.

**63**(5), 257–261 (1997) - 8.
Carbery, A., Wright, J.: Distributional and \(L^q\) norm inequalities for polynomials over convex bodies in \(\mathbb{R}^n\). Math. Res. Lett.

**3**(8), 233–248 (2001) - 9.
Ceccherini-Silberstein, T., Scarabotti, F., Tolli, F.: Harmonic Analysis on Finite Groups, Cambridge Studies in Advanced Mathematics, vol. 108. Cambridge University Press, Cambridge (2008)

- 10.
Ceccherini-Silberstein, T., Scarabotti, F., Tolli, F.: Representation Theory of the Symmetric Groups, Cambridge studies in Advanced Mathematics, vol. 121. Cambridge University Press, Cambridge (2010)

- 11.
Delsarte, P.: Association schemes and \(t\)-designs in regular semilattices. J. Comb. Theory Ser. A

**20**(2), 230–243 (1976) - 12.
Diaconis, P.: Group Representations in Probability and Statistics, Institute of Mathematical Statistics Lecture Notes—Monograph Series, vol. 11. Institute of Mathematical Statistics, Hayward (1988)

- 13.
Diaconis, P., Saloff-Coste, L.: Logarithmic Sobolev inequalities for finite Markov chains. Ann. Appl. Prob.

**6**(3), 695–750 (1996) - 14.
Dunkl, C.F.: A Krawtchouk polynomial addition theorem and wreath products of symmetric groups. Indiana Univ. Math. J.

**25**, 335–358 (1976) - 15.
Dunkl, C.F.: Orthogonal functions on some permutation groups. In: Relations Between Combinatorics and Other Parts of Mathematics, volume 34 of Proceedings of Symposia in Pure Mathematics, American Mathematcal Society, Providence, RI, pp. 129–147 (1979)

- 16.
Engel, K.: Sperner Theory, Encyclopedia of Mathematics and Its Applications, vol. 65. Cambridge University Press, Cambridge (1997)

- 17.
Filmus, Y.: An orthogonal basis for functions over a slice of the Boolean hypercube. Electron. J. Comb.

**23**(1), P1.23 (2016) - 18.
Filmus, Y., Kindler, G., Mossel, E., Wimmer, K.: Invariance principle on the slice. In: 31st Conference on Computational Complexity (2016)

- 19.
Håstad, J.: Almost optimal lower bounds for small depth circuits. In: Micali, S. (ed.) Randomness and Computation, Advances in Computing Research, vol. 5, pp. 143–170. JAI Press, Bingley (1989)

- 20.
Hoeffding, W.: Probability inequalities for sums of bounded random variables. J. Am. Stat. Assoc.

**58**(301), 13–30 (1963) - 21.
James, G.D.: A characteristic free approach to the representation theory of \(S_n\). J. Algebra

**46**, 430–450 (1977) - 22.
Keller, N., Klein, O.: A structure theorem for almost low-degree functions on the slice. Manuscript

- 23.
Kindler, G.: Property testing, PCP and Juntas. Ph.D. thesis, Tel-Aviv University (2002)

- 24.
Kindler, G., Safra, S.: Noise-resistant Boolean functions are juntas. Unpublished manuscript (2004)

- 25.
Lee, T., Prakash, A., de Wolf, R., Yuen, H.: On the sum-of-squares degree of symmetric quadratic functions. In: Proceedings of the 31st Conference on Computational Complexity (CCC 2016), pp. 17:1–17:31 (2016)

- 26.
Lee, T.-Y., Yau, H.-T.: Logarithmic Sobolev inequality for some models of random walks. Ann. Prob.

**26**(4), 1855–1873 (1998) - 27.
Linial, N., Mansour, Y., Nisan, N.: Constant depth circuits, fourier transform and learnability. J. ACM

**40**(3), 607–620 (1993) - 28.
Lyons, T.J., Zhang, T.S.: Decomposition of Dirichlet processes and its application. Ann. Probab.

**22**(1), 494–524 (1994) - 29.
Mossel, E., O’Donnell, R., Oleszkiewicz, K.: Noise stability of functions with low influences: invariance and optimality. Ann. Math.

**171**, 295–341 (2010) - 30.
Naor, A., Peres, Y., Schramm, O., Sheffield, S.: Markov chains in smooth Banach spaces and Gromov-hyperbolic metric spaces. Duke Math. J.

**134**(1), 165–197 (2006) - 31.
O’Donnell, R.: Analysis of Boolean Functions. Cambridge University Press, Cambridge (2014)

- 32.
O’Donnell, R., Wimmer, K.: Approximation by DNF: Examples and counterexamples, Automata, Languages and Programming, Lecture Notes in Computer Science, vol. 4596, pp. 195–206. Springer, Berlin (2007)

- 33.
Oleszkiewicz, K.: On a nonsymmetric version of the Khinchine–Kahane inequality. Prog. Probab.

**56**, 157–168 (2003) - 34.
Raghavendra, P., Weitz, B.: On the bit complexity of sum-of-squares proofs. arXiv:1702.05139 (2017)

- 35.
Sagan, B.E.: The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions, Graduate Texts in Mathematics, vol. 203. Springer, New York (2001)

- 36.
Srinivasan, M.K.: Symmetric chains, Gelfand–Tsetlin chains, and the Terwilliger algebra of the binary Hamming scheme. J. Algebra Comb.

**34**(2), 301–322 (2011) - 37.
Stanley, R.P.: Variations on differential posets. IMA Vol. Math. Appl.

**19**, 145–165 (1990) - 38.
Stanton, D.: Harmonics on posets. J. Comb. Theory Ser. A

**40**(1), 136–149 (1985) - 39.
Tal, A.: Tight bounds on the Fourier spectrum of \(AC^0\). Manuscript (2017)

- 40.
Talagrand, M.: On Russo’s approximate zero-one law. Ann. Probab.

**22**(3), 1576–1587 (1994) - 41.
Turner, L.R.: Inverse of the Vandermonde matrix with applications. Technical Report NASA TN D-3547, Lewis Research Center, NASA, Cleveland, Ohio (1966)

- 42.
Wimmer, K.: Low influence functions over slices of the Boolean hypercube depend on few coordinates. In: Conference on Computational Complexity (CCC 2014), pp. 120–131 (2014)

## 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.

## Author information

### Affiliations

### Corresponding author

## Additional information

### Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Rights and permissions

## About this article

### Cite this article

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

Received:

Revised:

Published:

Issue Date:

### Keywords

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

### Mathematics Subject Classification

- 60F17
- 60G99
- 05E30