Cutoff for non-negatively curved diffusions

TL;DR

This work resolves the cutoff problem for a broad class of Markov diffusions with non-negative Bakry–Émery curvature by proving a universal bound on the mixing-window width: $w_{\textsc{mix}}(\varepsilon)\le \frac{3}{\lambda \varepsilon^3}+3\sqrt{\frac{t_{\textsc{mix}}(1-\varepsilon)}{\lambda \varepsilon^3}}$, where $\lambda$ is the spectral gap. Consequently, a cutoff occurs whenever the product condition $\lambda t_{\textsc{mix}}(\varepsilon)\to\infty$, and the result extends to positive curvature, worst-case initial data, Euclidean and manifold settings, including Langevin diffusions in convex potentials. The proof hinges on a new differential inequality linking entropy and varentropy under CD$(0,\infty)$, together with a Pinsker-type relation and a spectral-gap-based mixing-time bound, yielding a simple, unifying approach that subsumes many model-specific cutoff proofs. Overall, the paper unifies and broadens cutoff phenomena for diffusions, providing a practical criterion to verify abrupt convergence across diverse continuous-state processes.

Abstract

We resolve the long-standing problem of elucidating the cutoff phenomenon for a vast and important class of Markov processes, namely Markov diffusions with non-negative Bakry-Émery curvature. More precisely, we prove that any sequence of non-negatively curved diffusions exhibits cutoff in total variation as soon as the product condition is satisfied. Our result holds in Euclidean spaces as well as on Riemannian manifolds, and for arbitrary non-random initial conditions. It vastly simplifies, unifies and generalizes a number of isolated works that have established cutoff through a delicate and model-dependent analysis of mixing times. The proof is elementary: we exploit a new simple differential relation between varentropy and entropy to produce a quantitative bound on the width of the mixing window.

Figures (1)

• Figure 1: A typical plot of the distance to equilibrium $t\mapsto {\textsc{tv}}(X_t)$. As the ratio $\frac{\mathrm{t}_{\textsc{mix}}(1-\varepsilon)}{\mathrm{t}_{\textsc{mix}}(\varepsilon)}$ approaches $1$, the transition to equilibrium becomes abrupt (cutoff).

Theorems & Definitions (11)

• Example 1: Langevin diffusion in a convex potential
• Example 2: Langevin diffusion on a weighted Riemannian manifold
• Theorem 1: Mixing window
• Corollary 1: Cutoff
• Theorem 2: Mixing window under positive curvature
• Lemma 1: Reverse Pinsker's inequality
• Lemma 2: Mixing-time estimate
• Lemma 3: Information-theoretic differential inequality
• proof
• proof : Proof of Theorem \ref{['th:main']}