Log-Sobolev inequality for the multislice, with applications

Abstract

Let κ∈N+ℓ satisfy κ1+⋯+κℓ=n, and let Uκ denote the multislice of all strings u∈[ℓ]n having exactly κi coordinates equal to i, for all i∈[ℓ]. Consider the Markov chain on Uκ where a step is a random transposition of two coordinates of u. We show that the log-Sobolev constant ϱκ for the chain satisfies ϱκ−1≤n⋅∑i=1ℓ12log2(4n∕κi), 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.