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A Log-Sobolev Inequality for the Multislice, with Applications

Authors Yuval Filmus, Ryan O'Donnell, Xinyu Wu

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Author Details

Yuval Filmus
  • Department of Computer Science, Technion - Israel Institute of Technology, Haifa, Israel
Ryan O'Donnell
  • Computer Science Department, Carnegie Mellon University, Pittsburgh, PA, USA
Xinyu Wu
  • Computer Science Department, Carnegie Mellon University, Pittsburgh, PA, USA

Funding

  • Filmus, Yuval: Taub Fellow - supported by the Taub Foundations. The research was funded by ISF grant 1337/16.
  • O'Donnell, Ryan: 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).

Cite AsGet BibTex

Yuval Filmus, Ryan O'Donnell, and Xinyu Wu. A Log-Sobolev Inequality for the Multislice, with Applications. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 34:1-34:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.ITCS.2019.34

Abstract

Let kappa in N_+^l satisfy kappa_1 + *s + kappa_l = n, and let U_kappa denote the multislice of all strings u in [l]^n having exactly kappa_i coordinates equal to i, for all i in [l]. Consider the Markov chain on U_kappa where a step is a random transposition of two coordinates of u. We show that the log-Sobolev constant rho_kappa for the chain satisfies rho_kappa^{-1} <= n * sum_{i=1}^l 1/2 log_2(4n/kappa_i), which is sharp up to constants whenever l 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.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Markov processes
Keywords
  • log-Sobolev inequality
  • small-set expansion
  • conductance
  • hypercontractivity
  • Fourier analysis
  • representation theory
  • Markov chains
  • combinatorics

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