A Markov chain converges to its stationary distribution gradually — or so intuition suggests. The cutoff phenomenon contradicts this: many chains remain far from equilibrium for a long time, then snap to stationarity within a narrow window. The transition from “unmixed” to “mixed” is not gradual but abrupt. The chain waits, then switches.
Establishing cutoff for specific chains has required intricate, model-by-model analysis. Each proof technique was tailored to the chain's structure — random walks on groups needed representation theory, card shuffles needed coupling arguments, interacting particles needed hydrodynamic limits. The phenomenon appeared universal, but each demonstration was bespoke.
The paper proves cutoff universally for a broad class: any diffusion process with non-negative Bakry-Émery curvature, in arbitrary dimension, on Euclidean space or Riemannian manifolds, with any initial condition. The only requirement is a “product condition” — the spectral gap times the mixing time diverges. Under non-negative curvature, this single condition guarantees cutoff.
The proof introduces a new tool: a differential inequality relating varentropy (the variance of the log-density) to entropy. This single relation, combined with the curvature bound, controls the entire mixing profile. The intricate analyses that previous work required for individual chains all collapse into consequences of this one inequality.
The structural point: curvature controls convergence shape, not just convergence rate. Non-negative curvature doesn't just make the chain mix — it makes the chain mix sharply. The geometry of the state space determines not whether the chain reaches equilibrium but how it reaches equilibrium: gradually or all at once.