On cutoff via rigidity for high dimensional curved diffusions

TL;DR

This paper demonstrates a cutoff phenomenon in high-dimensional curved diffusions, including non-Gaussian cases, at a critical time related to the spectral gap, using functional inequalities and spectral rigidity concepts.

Contribution

It establishes the occurrence of cutoff in high-dimensional curved diffusions at a specific time, extending the understanding beyond Gaussian cases with new spectral and functional inequality techniques.

Findings

Cutoff occurs at time log(dimension)/(2 * spectral gap)

Cutoff holds across Wasserstein, total variation, entropy, and Fisher information

Extension to Riemannian manifolds and new curvature product condition introduced

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 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 progress on the product condition for non-negatively curved diffusions leads us to introduce a new curvature product condition.