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Concentration of measure on spheres and related manifolds

Friedrich Götze & Holger Sambale

Electronic Journal of Probability2026https://doi.org/10.1214/26-ejp1492article
ABDC A
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0.50

Abstract

We study various generalizations of concentration of measure on the unit sphere, in particular by means of log-Sobolev inequalities. First, we show Sudakov-type concentration results and local semicircular laws for weighted random matrices. A further branch addresses higher order concentration (i. e., concentration for non-Lipschitz functions which have bounded derivatives of higher order) for ℓpn-spheres. This is based on a type of generalized log-Sobolev inqualities referred to as LSq-inequalities. More generally, we prove higher order concentration bounds for probability measures on Rn which satisfy an LSq-inequality. Finally, we derive concentration bounds for sequences of smooth symmetric functions on the Euclidean sphere which are closely related to Edgeworth-type expansions.

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https://doi.org/https://doi.org/10.1214/26-ejp1492

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@article{friedrich2026,
  title        = {{Concentration of measure on spheres and related manifolds}},
  author       = {Friedrich Götze & Holger Sambale},
  journal      = {Electronic Journal of Probability},
  year         = {2026},
  doi          = {https://doi.org/https://doi.org/10.1214/26-ejp1492},
}

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Concentration of measure on spheres and related manifolds

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