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2022 Log-Sobolev inequality for the multislice, with applications
Yuval Filmus, Ryan O’Donnell, Xinyu Wu
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Electron. J. Probab. 27: 1-30 (2022). DOI: 10.1214/22-EJP749

Abstract

Let κN+ satisfy κ1++κ=n, and let Uκ denote the multislice of all strings u[]n having exactly κi coordinates equal to i, for all i[]. Consider the Markov chain on Uκ where a step is a random transposition of two coordinates of u. We show that the log-Sobolev constant ϱκ for the chain satisfies

ϱκ1ni=112log2(4nκi),

which is sharp up to constants whenever is constant. From this, we derive some consequences for small-set expansion and isoperimetry in the multislice, including a KKL Theorem, a Kruskal–Katona Theorem for the multislice, a Friedgut Junta Theorem, and a Nisan–Szegedy Theorem.

Funding Statement

Y. Filmus (Taub Fellow) was supported by the Taub Foundations. The research was funded by ISF grant 1337/16. R. O’Donnell was supported by NSF grant CCF-1717606. This material is based upon work supported by the National Science Foundation under grant numbers listed above. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation (NSF).

Citation

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Yuval Filmus. Ryan O’Donnell. Xinyu Wu. "Log-Sobolev inequality for the multislice, with applications." Electron. J. Probab. 27 1 - 30, 2022. https://doi.org/10.1214/22-EJP749

Information

Received: 10 September 2018; Accepted: 30 January 2022; Published: 2022
First available in Project Euclid: 2 March 2022

arXiv: 1809.03546
Digital Object Identifier: 10.1214/22-EJP749

Subjects:
Primary: 05E18 , 60J10 , 68R05

Keywords: combinatorics , conductance , Fourier analysis , hypercontractivity , Log-Sobolev inequality , Markov chains , representation theory , small-set expansion

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