Log-Sobolev inequality for the multislice, with applications

Abstract

Let $\mathrm{\kappa }\in {\mathbb{N}}_{+}^{\ell }$ satisfy ${\mathrm{\kappa }}_1+\cdots +{\mathrm{\kappa }}_{\ell }=n$, and let ${\mathcal{U}}_{\mathrm{\kappa }}$ denote the multislice of all strings $u\in {[\ell ]}^n$ having exactly ${\mathrm{\kappa }}_i$ coordinates equal to i, for all $i\in [\ell ]$. Consider the Markov chain on ${\mathcal{U}}_{\mathrm{\kappa }}$ where a step is a random transposition of two coordinates of u. We show that the log-Sobolev constant ${\mathit{\varrho }}_{\mathrm{\kappa }}$ for the chain satisfies

$${\mathit{\varrho }}_{\mathrm{\kappa }}^{-1}\le n\cdot \sum _{i=1}^{\ell }\frac{1}{2}{log}_2(4n∕{\mathrm{\kappa }}_i),$$

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.