A log-Sobolev inequality for the multislice, with applications

Let $\kappa \in {\mathbb{N}}_{+}^{\mathrm{\ell }}$ satisfy $\kappa_1 + \dots + \kappa_\ell = n$andlet$$\mathcal{U}$\kappa$denotethe\mathrm{"}multislice\mathrm{"}ofallstrings$u$in$[\ell]^n$havingexactly$\kappa_i$coordinatesequalto$i$,forall$i \in [\ell]$\mathrm{.}ConsidertheMarkovchainon$$\mathcal{U}$\kappa$, where a step is a random transposition of two coordinates of $u$. We show that the log-Sobolev constant ${\rho }_{\kappa }$ for the chain satisfies $$({\rho }_{\kappa }{)}^{-1}\le n\sum _{i=1}^{\mathrm{\ell }}\frac{1}{2}{\log }_2(4n\mathrm{/}{\kappa }_i),$$ which is sharp up to constants whenever $\mathrm{\ell }$ 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.