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A log-Sobolev inequality for the multislice, with applications

Yuval Filmus, Ryan O'Donnell and Xinyu Wu
ITCS 2019

The multislice is a generalization of the slice. Whereas the slice consists of all vectors in $\{0,1\}^n$ of fixed Hamming weight, the multislice consists of all vectors in $[\ell]^n$ of fixed histogram.

The log-Sobolev inequality is a fundamental inequality in Boolean Function Analysis, equivalent to hypercontractivity. Lee and Yau determined (up to a constant factor) the optimal constant in the log-Sobolev inequality for the slice. We give a clear exposition of their argument, and extend it to the multislice.

As applications, we derive versions of the Kahn–Kalai–Linial, Friedgut’s junta, Kruskal–Katona, and Nisan–Szegedy theorems.

Our log-Sobolev inequality is tight only for constant $\ell$. Justin Salez has since proved a tight log-Sobolev inequality for all multislices.

BibTeX

@inproceedings{FOW2019,
 author = {Yuval Filmus and Ryan O'Donnell and Xinyu Wu},
 title = {A log-{S}obolev inequality for the multislice, with applications},
 booktitle = {Proceedings of the 10th Innovations in Theoretical Computer Science conference (ITCS'19)},
 year = {2019}
}
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