On cutoff via rigidity for high dimensional curved diffusions

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.

💡 Research Summary

The paper investigates abrupt convergence (“cutoff”) for high‑dimensional overdamped Langevin diffusions whose curvature equals the spectral gap. The authors consider the stochastic differential equation
(dX_t = -\nabla V(X_t),dt + \sqrt{2},dB_t)
on (\mathbb R^d) with a strictly convex, (C^2) potential (V) such that (\mu(dx)=e^{-V(x)}dx) is a probability measure. The generator (L=\Delta-\nabla V\cdot\nabla) is symmetric in (L^2(\mu)) and its spectrum consists of a simple eigenvalue at 0 (constants) and the rest bounded above by (-\lambda_1), where (\lambda_1>0) is the spectral gap.

A central hypothesis is the curvature condition
(\text{Hess}(V)(x) \succeq \lambda_1 I_d) for all (x), i.e. the Bakry–Émery curvature is at least (\lambda_1). This is equivalent to the curvature‑dimension inequality (CD(\lambda_1,\infty)) and guarantees that (\mu) is (\lambda_1)-log‑concave (strongly convex in the sense of functional inequalities).

Main quantitative result (Theorem 1.1).
For any non‑empty set of initial points (S\subset\mathbb R^d) the Wasserstein‑2 distance satisfies
\