A log-Sobolev inequality for the multislice, with applications

Yuval Filmus, Ryan O'Donnell, Xinyu Wu

Let $\kappa \in {\mathbb{N}}_{+}^{\ell }$ satisfy ${\kappa }_1+\cdots +{\kappa }_{\ell }=n$ and let ${\mathcal{U}}_{\kappa }$ denote the "multislice" of all strings $u$ in $[\ell {]}^n$ having exactly ${\kappa }_i$ coordinates eq…