A Log-Sobolev Inequality for the Multislice, with Applications

Yuval Filmus, Ryan O'Donnell, Xinyu Wu, Michael Wagner
2018 Innovations in Theoretical Computer Science  
Let κ ∈ N + satisfy κ 1 + • • • + κ = n, and let U κ denote the multislice of all strings u ∈ [ ] n having exactly κ i coordinates equal to i, for all i ∈ [ ]. Consider the Markov chain on U κ where a step is a random transposition of two coordinates of u. We show that the log-Sobolev constant κ for the chain satisfies 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
more » ... Kruskal-Katona Theorem for the multislice, a Friedgut Junta Theorem, and a Nisan-Szegedy Theorem.
doi:10.4230/lipics.itcs.2019.34 dblp:conf/innovations/FilmusOW19 fatcat:bkvdwesdobe7nab3ajopcduo4q