On cutoff via rigidity for high dimensional curved diffusions

Djalil Chafaï; Max Fathi

doi:10.5802/crmath.776

Research article - Functional analysis, Probability theory

On cutoff via rigidity for high dimensional curved diffusions

Djalil Chafaï ^1,²; Max Fathi ^1,³

¹ DMA, École normale supérieure, 45 rue d’Ulm, 75230 Paris, France
² CEREMADE, Université Paris-Dauphine, Place du Maréchal de Lattre de Tassigny, 75775 Paris, France
³ Université Paris Cité and Sorbonne Université, CNRS, Laboratoire Jacques-Louis Lions and Laboratoire de Probabilités, Statistique et Modélisation, 75013 Paris, France

Comptes Rendus. Mathématique, Volume 363 (2025), pp. 1103-1121

Abstract (Original language)
Résumé

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 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.

Nous considérons des diffusions de Langevin sur-amorties dans l’espace euclidien, avec une courbure égale au trou spectral. Ceci inclut le processus d’Ornstein–Uhlenbeck ainsi que des extensions non gaussiennes et non-produit avec interaction convexe, telles que le processus de Dyson issu de la théorie des matrices aléatoires. Nous montrons qu’un phénomène de convergence abrupte vers l’équilibre se produit en grande dimension, à un temps critique égal au logarithme de la dimension divisé par deux fois le trou spectral. Cela a lieu pour la distance de Wasserstein, la variation totale, l’entropie relative et l’information de Fisher. Une observation clé est une relation à une rigidité spectrale, liée à la présence d’un facteur gaussien. Une nouveauté est l’utilisation extensive d’inégalités fonctionnelles, même pour la régularisation en temps court, et la réduction à Wasserstein. Les preuves sont courtes et conceptuelles. Puisque la condition produit est satisfaite, une coupure $Lp$ est valable pour tout $p$. Nous discutons également d’une extension naturelle aux variétés riemanniennes, d’un lien avec les estimations de gradient logarithmique en temps court pour le noyau de chaleur, et nous nous interrogeons sur la stabilité par perturbation. Enfin, au-delà de la rigidité, mais toujours pour les diffusions, une discussion autour des progrès récents sur la condition de produit pour les diffusions à courbure positive nous conduit à introduire une nouvelle condition de produit de courbure.

Received: 2024-12-20
Revised: 2025-06-04
Accepted: 2025-07-04
Published online: 2025-10-13

DOI: 10.5802/crmath.776

Classification: 60J60, 58J65, 47D07, 53C21, 58J50
Keywords: Markov diffusion process, curvature-dimension inequality, spectral gap, ergodicity and cutoff, Wasserstein distance, relative entropy, total variation, Fisher information
Mots-clés : Processus de diffusions markovien, inégalité de courbure dimension, trou spectral, ergodicity et convergence abrupte, distance de Wasserstein, entropie relative, information de Fisher

Author's affiliations:

Djalil Chafaï ^{1, 2}; Max Fathi ^{1, 3}

¹ DMA, École normale supérieure, 45 rue d’Ulm, 75230 Paris, France
² CEREMADE, Université Paris-Dauphine, Place du Maréchal de Lattre de Tassigny, 75775 Paris, France
³ Université Paris Cité and Sorbonne Université, CNRS, Laboratoire Jacques-Louis Lions and Laboratoire de Probabilités, Statistique et Modélisation, 75013 Paris, France

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     title = {On cutoff via rigidity for high dimensional curved diffusions},
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     volume = {363},
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     language = {en},
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Djalil Chafaï; Max Fathi. On cutoff via rigidity for high dimensional curved diffusions. Comptes Rendus. Mathématique, Volume 363 (2025), pp. 1103-1121. doi: 10.5802/crmath.776

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