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On cutoff via rigidity for high dimensional curved diffusions

Djalil Chafaï, Max Fathi

TL;DR

The paper analyzes high-dimensional overdamped Langevin diffusions with curvature bounded below by the spectral gap and proves a sharp cutoff at $t_* = \frac{\log d}{2\lambda_1}$ across Wasserstein distance, total variation, relative entropy, and Fisher information. Central to the results is a rigidity phenomenon that yields a Gaussian factor in the invariant measure, allowing a reduction to a lower-dimensional OU process and enabling explicit, dimension-dependent mixing times. The authors establish a product condition and show $L^p$ cutoff for all $p\ge 1$ in curved settings, with extensions to weighted Riemannian manifolds via Bakry--Émery calculus and a Gaussian–factorization splitting theorem. They also explore stability questions, short-time Fisher information bounds, and normalization/temperature invariance, situating their results within the curvature-dimension framework and linking rigidity to spectral properties. Overall, the work provides explicit high-dimensional mixing time scales for a broad class of diffusions, including non-Gaussian and non-product interactions such as Dyson-type models, and lays groundwork for further geometric extensions.

Abstract

We consider overdamped Langevin diffusions in Euclidean space, with curvature equal to the spectral gap. This includes the Ornstein-Uhlenbeck process as well as non Gaussian and non product extensions with convex interaction, such as the Dyson process from random matrix theory. We show that a cutoff phenomenon or abrupt convergence to equilibrium occurs in high-dimension, at a critical time equal to the logarithm of the dimension divided by twice the spectral gap. This cutoff holds for Wasserstein distance, total variation, relative entropy, and Fisher information. A key observation is a relation to a spectral rigidity, linked to the presence of a Gaussian factor. A novelty is the extensive usage of functional inequalities, even for short-time regularization, and the reduction to Wasserstein. The proofs are short and conceptual. Since the product condition is satisfied, an Lp cutoff hold for all p. We moreover discuss a natural extension to Riemannian manifolds, a link with logarithmic gradient estimates in short-time for the heat kernel, and ask about stability by perturbation. Finally, beyond rigidity but still for diffusions, a discussion around the recent progresses on the product condition for nonnegatively curved diffusions leads us to introduce a new curvature product condition.

On cutoff via rigidity for high dimensional curved diffusions

TL;DR

The paper analyzes high-dimensional overdamped Langevin diffusions with curvature bounded below by the spectral gap and proves a sharp cutoff at across Wasserstein distance, total variation, relative entropy, and Fisher information. Central to the results is a rigidity phenomenon that yields a Gaussian factor in the invariant measure, allowing a reduction to a lower-dimensional OU process and enabling explicit, dimension-dependent mixing times. The authors establish a product condition and show cutoff for all in curved settings, with extensions to weighted Riemannian manifolds via Bakry--Émery calculus and a Gaussian–factorization splitting theorem. They also explore stability questions, short-time Fisher information bounds, and normalization/temperature invariance, situating their results within the curvature-dimension framework and linking rigidity to spectral properties. Overall, the work provides explicit high-dimensional mixing time scales for a broad class of diffusions, including non-Gaussian and non-product interactions such as Dyson-type models, and lays groundwork for further geometric extensions.

Abstract

We consider overdamped Langevin diffusions in Euclidean space, with curvature equal to the spectral gap. This includes the Ornstein-Uhlenbeck process as well as non Gaussian and non product extensions with convex interaction, such as the Dyson process from random matrix theory. We show that a cutoff phenomenon or abrupt convergence to equilibrium occurs in high-dimension, at a critical time equal to the logarithm of the dimension divided by twice the spectral gap. This cutoff holds for Wasserstein distance, total variation, relative entropy, and Fisher information. A key observation is a relation to a spectral rigidity, linked to the presence of a Gaussian factor. A novelty is the extensive usage of functional inequalities, even for short-time regularization, and the reduction to Wasserstein. The proofs are short and conceptual. Since the product condition is satisfied, an Lp cutoff hold for all p. We moreover discuss a natural extension to Riemannian manifolds, a link with logarithmic gradient estimates in short-time for the heat kernel, and ask about stability by perturbation. Finally, beyond rigidity but still for diffusions, a discussion around the recent progresses on the product condition for nonnegatively curved diffusions leads us to introduce a new curvature product condition.

Paper Structure

This paper contains 22 sections, 11 theorems, 95 equations.

Table of Contents

  1. Introduction and main results
  2. Diffusions on Euclidean spaces, with convex potential
  3. Cutoff for high dimensional curved diffusions
  4. Product condition and Lp cutoff at the mixing time
  5. Positively curved diffusions, curvature product condition, and cutoff
  6. Rigidity and Gaussian factorization for curved diffusions
  7. Riemannian manifolds
  8. About stability
  9. Further comments
  10. Fisher information and the heat kernel
  11. Optimal constants
  12. Normalization
  13. Temperature
  14. Proofs
  15. Preliminaries on rigidity
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let ${(X_t)}_{t\geq0}$ be the process eq:eds, with potential $V$, spectral gap $\lambda_1$, and invariant law $\mu=\mathrm{e}^{-V}$. If, for all $x\in\mathbb{R}^d$, and as quadratic forms, then for all non-empty set of initial conditions $S\subset\mathbb{R}^d$, with, denoting $E_1$ the eigenspace of $-\mathcal{L}$ associated to $\lambda_1$, and $k_1=\dim(E_1)$, where the supremum runs over the

Theorems & Definitions (17)

  • Theorem 1.1: Main Wasserstein estimate
  • Remark 1.2: Dimensions
  • Corollary 1.3: Wasserstein cutoff
  • Corollary 1.4: I, H, and TV cutoffs
  • Theorem 1.5: Boltzmann -- Gibbs probability measure with convex interaction
  • Corollary 1.6: Cutoff for Langevin with convex interactions
  • Theorem 1.7: Cutoff for positively curved diffusions with curvature product condition
  • Definition 1.8: Multi-eigenfunction
  • Theorem 1.9: Wasserstein estimate on weighted Riemannian manifold
  • Theorem 3.1: Gaussian factorization in the Euclidean space
  • ...and 7 more