Metastability and Sharp Propagation-of-Chaos Thresholds for Mean-Field Interacting Diffusions on the Circle

Abstract

We study metastability and long-time propagation-of-chaos for mean-field interacting diffusions on the circle with multiple local free-energy minima. Working modulo rotations, we identify sharp exponential time scales for metastability and the validity and breakdown of the McKean-Vlasov approximation. Assuming finitely many critical rotation orbits and distinct local-minimum energies, we show that, for initial data in the attraction basin of a nonglobal minimum, metastability and propagation-of-chaos hold uniformly over deterministic times up to $e^{N(D-\varepsilon)}$ and fail at every deterministic sequence growing faster than $e^{N(D+\varepsilon)}$. Here $N$ is the particle number and $D$ the depth of the maximal Freidlin-Wentzell energy cycle having that minimum as its bottom. The same threshold governs running averages of the squared approximation error in a stronger pathwise sense, with corresponding metastability and propagation-of-chaos bounds for suitably aligned time-averaged empirical measures. When the global-minimum orbit is unique, we establish uniform-in-time propagation-of-chaos in its attraction basin without assuming finitely many critical orbits. The proofs develop Freidlin-Wentzell cycle analysis for the empirical measure process on the rotation quotient. The key steps extend Dawson and Gärtner's fixed-horizon large deviation estimates to excursions of unbounded duration between metastable wells, yielding sharp cycle-exit estimates and localization at deterministic times. We give a checkable Fourier criterion for finiteness of the critical orbits and explicit examples with multiple wells, and discuss applications to the noisy Kuramoto model and conditional consequences for noisy transformer interactions. We also investigate how phase transitions in the free-energy landscape, as noise strength varies, affect metastability and propagation-of-chaos.