The local product condition implies cutoff

Abstract

In the theory of mixing times, a famously wrong conjecture predicts that a sequence of Markov processes exhibits cutoff as soon as the product of their Poincaré constant and mixing time diverges. We prove that this statement becomes correct once the Poincaré constant $γ$ is replaced with its natural non-equilibrium refinement, which we denote by $γ_\star$. More precisely, we show that the width of the mixing window of any Markov process is $O(1/γ_\star)$. This estimate is sharp, and universal up to standard regularity assumptions: it holds on finite and infinite state spaces and from any initial condition, and it does not require reversibility, nor any kind of a chain rule. In addition, for deterministic initialization we show that $γ_\star\geκ$, where $κ$ is the Bakry-Émery curvature, making our result broadly applicable. Finally, our proof is short and self-contained: we simply follow the classical idea of replacing the total variation distance by the more tractable $χ^2$-divergence, but with the crucial novelty that the reference measure evolves in time, instead of being the equilibrium law.