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

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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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