The local product condition implies cutoff

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 Poincar\'e constant and mixing time diverges. We prove that this statement becomes correct once the Poincar\'e constant $\gamma$ is replaced with its natural non-equilibrium refinement, which we denote by $\gamma_\star$. More precisely, we show that the width of the mixing window of any Markov process is $O(1/\gamma_\star)$. 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 $\gamma_\star\ge\kappa$, where $\kappa$ is the Bakry-\'Emery 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 $\chi^2$-divergence, but with the crucial novelty that the reference measure evolves in time, instead of being the equilibrium law.