Year: 2025
Type: article
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 L p cutoff holds 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.
Source: Comptes Rendus Mathématique
DOI: 10.5802/crmath.776
Authors: Djalil Chafaï, Max Fathi
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: diamond
Funder: Agence Nationale de la Recherche
OpenAlex ID: W4415104267