Log-Sobolev inequality for the multislice, with applications

Let Kappa is an element of N`+ satisfy Kappa 1 + middot middot middot + Kappa`= n, and let U Kappa denote the multislice of all strings u is an element of [B]n having exactly Kappa i coordinates equal to i, for all i is an element of [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 p Kappa for the chain satisfies sigma(-1)(K) <= n center dot Sigma(i=1)log(2)(4n/Kappa(i)), which is sharp up to constants whenever B 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.