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Noise Stability on the Hypercube: Order 7/8

  • Lei Yu
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Abstract

For a balanced Boolean function f on the discrete hypercube, we study the α-stability 𝔼[(Tρf)α], where Tρ is the Boolean noise operator and 0<α<1. Using the asymmetric Hellinger kernel estimates recently established by OpenAI, we prove that dictators minimize the 7/8-stability in every dimension and at every correlation 0≤ρ≤1. At interior correlations they are the only minimizers. As a consequence, dictators minimize the α-stability for every 7/8≤α<1, maximize the α-stability for every 1<α<2, and maximize the one-sided Shannon entropy stability 𝔼[Tρflog Tρf]. This resolves a conjecture of Li and Médard and strengthens the Courtade--Kumar conjecture in the balanced setting.

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Cite

BibTeXRIS
BibTeX
@misc{yu2026noise,
  title = {{Noise Stability on the Hypercube: Order $7/8$}},
  author = {Lei Yu},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.00868},
  eprint = {2611.00868},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.00868},
}
RIS
TY  - UNPB
TI  - Noise Stability on the Hypercube: Order 7/8
AU  - Lei Yu
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.00868
UR  - https://arxiv.org/abs/2611.00868
AB  - For a balanced Boolean function f on the discrete hypercube, we study the α-stability 𝔼[(T_ρf)^α], where T_ρ is the Boolean noise operator and 0<α<1. Using the asymmetric Hellinger kernel estimates recently established by OpenAI, we prove that dictators minimize the 7/8-stability in every dimension and at every correlation 0≤ρ≤1. At interior correlations they are the only minimizers. As a consequence, dictators minimize the α-stability for every 7/8≤α<1, maximize the α-stability for every 1<α<2, and maximize the one-sided Shannon entropy stability 𝔼[T_ρflog T_ρf]. This resolves a conjecture of Li and Médard and strengthens the Courtade--Kumar conjecture in the balanced setting.
ER  -