This paper shows that a small, natural change to a classical quantity fixes a long-standing guess about when Markov processes show an abrupt transition to equilibrium, known as cutoff. The old conjecture said that cutoff happens as soon as the product of the Poincaré constant and the mixing time grows without bound. The authors prove a correct version of this statement by replacing the usual Poincaré constant with a time-dependent, non-equilibrium version they call the local Poincaré constant (denoted γ★).

Roughly speaking, mixing time measures how long a process takes to forget its start and look like its long-term distribution. Cutoff means this change happens very suddenly as a system parameter (often the size) grows. The paper proves that the width of the mixing window — the time interval in which the process moves from “not mixed” to “well mixed” — is bounded by a constant times 1/γ★. Concretely, for any target precision ε in (0,1/2), the authors give the explicit bound w_mix(ε) ≤ 32·(1/γ★)·log(1/ε). This makes the product condition correct if one uses γ★ instead of the usual Poincaré constant γ.

What the authors did and how the proof works. They define γ★ as the worst (smallest) Poincaré constant observed along the process over time. The main technical idea is to control a tractable measure of distance called the χ2-divergence, instead of the usual total-variation distance, and to let the reference measure evolve in time rather than fixing it at equilibrium. This change yields an exponential decay bound for χ2 that is governed by γ★, which in turn gives the window-width estimate above. The proof is short and self-contained and does not require the process to be reversible or to satisfy any special chain rule.

Why this matters. The result gives a clean and broadly applicable criterion for cutoff that works on both finite and infinite state spaces and from any initial condition. It unifies and improves earlier results that required restrictive assumptions. The paper also connects γ★ to a familiar geometric quantity: when the chain starts from a fixed state (deterministic initialization), they show γ★ is at least the Bakry–Émery curvature κ. As a result, positive curvature immediately implies a small mixing window and hence cutoff with window size O(1/κ·log(1/ε)). The authors also give a simple example — the random walk on the n-dimensional hypercube — where γ★ equals the classical Poincaré constant and the bound is sharp.