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

@misc{banerjee2026metastability,
  title = {{Metastability and Sharp Propagation-of-Chaos Thresholds for Mean-Field Interacting Diffusions on the Circle}},
  author = {Sayan Banerjee},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.08755},
  eprint = {2610.08755},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.08755},
}

TY  - UNPB
TI  - Metastability and Sharp Propagation-of-Chaos Thresholds for Mean-Field Interacting Diffusions on the Circle
AU  - Sayan Banerjee
PY  - 2026
DA  - 2026/10/14
PB  - arXiv
DO  - 10.48550/arxiv.2610.08755
UR  - https://arxiv.org/abs/2610.08755
AB  - 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−ε) and fail at every deterministic sequence growing faster than e^N(D+ε). 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.
ER  -