Log-Sobolev inequality for the multislice, with applications

Let κ∈N^ℓ ₊ satisfy κ₁+⋯+ κ_ℓ=n, and let U_κ 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 U_κ where a step is a random transposition of two coordinates of u. We show that the log-Sobolev constant ϱκ for the chain satisfies (formula presented), 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.