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 ${\hat{I}}^3$ is replaced with its natural non-equilibrium refinement, which we denote by ${\hat{I}}_{\star }^3$. More precisely, we show that the width of the mixing window of any Markov process is $O(1/{\hat{I}}_{\star }^3)$. 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 ${\hat{I}}_{\star }^3\ge \hat{I}º$, where $\hat{I}º$ 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 $\ddot{I}{‡}^2$-divergence, but with the crucial novelty that the reference measure evolves in time, instead of being the equilibrium law.

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