Let satisfy , and let denote the multislice of all strings having exactly coordinates equal to i, for all . Consider the Markov chain on where a step is a random transposition of two coordinates of u. We show that the log-Sobolev constant for the chain satisfies
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.
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).
"Log-Sobolev inequality for the multislice, with applications." Electron. J. Probab. 27 1 - 30, 2022. https://doi.org/10.1214/22-EJP749