Mixing time under monotone censoring

Abstract

We prove that the lazy random walk on the discrete cube, censored to any increasing set of fixed positive density, mixes in time $O(n\log n)$, answering a question of Ding and Mossel. More precisely, for every nonempty increasing set $A\subseteq \{0,1{\}}^n$, \begin{equation} t_{\mathrm{mix}}(P) \le Kμ(A)^{-3}n\log(en). \label{eq:mixing-time-bound} \end{equation} where $\mu$ is uniform on the cube and $K$ is an absolute constant. The proof uses hypercontractivity on the ambient cube to strengthen Poincaré inequality on coordinate sections. A stopping-time occupation inequality for increasing sets converts the resulting local bound into a uniform bound on hitting times of large sets. See the appendix for a better estimates of the constant and the dependence on \(μ(A)\).

25 pages