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PreprintCC BY 4.0arXiv

Rapid mixing for Gibbs measures in Riemannian manifolds

  • Ángela Capel
  • Marco Castrillón-López
  • Sofyan Iblisdir
  • Angelo Lucia
  • Pablo Páez-Velasco
  • David Pérez-García
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Abstract

Langevin dynamics on Riemannian manifolds is analyzed. Conditions ensuring the existence of a suitable logarithmic Sobolev inequality (rapid mixing to the Gibbs measure) are identified. These conditions involve the curvature of the manifold, the inverse temperature, escaping directions from saddle points, and exclude barren plateaus and spurious local minima. We show that when these conditions are met, mixing times polynomial in the dimension of the manifold are achievable. This result is obtained through a relation between Langevin processes in the domain and in the image of a Riemannian submersion. Such a relation can be of independent interest.

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Cite

BibTeXRIS
BibTeX
@misc{capel2026rapid,
  title = {{Rapid mixing for Gibbs measures in Riemannian manifolds}},
  author = {Ángela Capel and Marco Castrillón-López and Sofyan Iblisdir and Angelo Lucia and Pablo Páez-Velasco and David Pérez-García},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2606.13453},
  eprint = {2606.13453},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.13453},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Rapid mixing for Gibbs measures in Riemannian manifolds
AU  - Ángela Capel
AU  - Marco Castrillón-López
AU  - Sofyan Iblisdir
AU  - Angelo Lucia
AU  - Pablo Páez-Velasco
AU  - David Pérez-García
PY  - 2026
DA  - 2026/09/17
PB  - arXiv
DO  - 10.48550/arxiv.2606.13453
UR  - https://arxiv.org/abs/2606.13453
AB  - Langevin dynamics on Riemannian manifolds is analyzed. Conditions ensuring the existence of a suitable logarithmic Sobolev inequality (rapid mixing to the Gibbs measure) are identified. These conditions involve the curvature of the manifold, the inverse temperature, escaping directions from saddle points, and exclude barren plateaus and spurious local minima. We show that when these conditions are met, mixing times polynomial in the dimension of the manifold are achievable. This result is obtained through a relation between Langevin processes in the domain and in the image of a Riemannian submersion. Such a relation can be of independent interest.
ER  -