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 $e^{N(D-\epsilon )}$ and fail at every deterministic sequence growing faster than $e^{N(D+\epsilon )}$. 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 the Freidlin-Wentzell cycle analysis for the empirical measure process and exploit Dawson and Gärtner's fixed-horizon large deviation estimates to analyze long excursions between metastable wells. We give a checkable Fourier criterion for finiteness of the critical orbits. We also investigate connections with phase transitions in the free-energy landscape, and discuss applications to the noisy Kuramoto and noisy transformer models. We provide, to our knowledge, the first sharp exponential-time analysis of the Dawson-Gärtner `tunnelling phenomenon' for general smooth interactions, with two-sided probability bounds and matching propagation-of-chaos thresholds.

arXiv commentMinor typos corrected. Some additional comparisons to existing work and acknowledgements added. 78 pages