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Random Hamiltonians II: A central limit theorem and the Hofer geometry of random walks — arXiv:2608.07673

Source: arXiv (en)
Original: https://arxiv.org/abs/2608.07673
License (as archived): CC0-1.0
Archived: 2026-10-19T09:07:06.604Z
SHA-256 of the archived text: 30bfc13b76bb5f29b56bda959c8f4e9109ab76c14e4ef82a0e2bb5ab834a06db
Stable URL of this page: https://metraipolis.com/documento/dc_fdb1df975017

How to cite

Random Hamiltonians II: A central limit theorem and the Hofer geometry of random walks — arXiv:2608.07673 — arXiv (en), https://arxiv.org/abs/2608.07673 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_fdb1df975017 (Metraipolis archive dc_fdb1df975017, sha256 30bfc13b76bb5f29b56bda959c8f4e9109ab76c14e4ef82a0e2bb5ab834a06db)

BibTeX

@misc{dcfdb1df975017,
  title = {Random Hamiltonians II: A central limit theorem and the Hofer geometry of random walks — arXiv:2608.07673},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2608.07673},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_fdb1df975017; sha256 30bfc13b76bb5f29b56bda959c8f4e9109ab76c14e4ef82a0e2bb5ab834a06db},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2608.07673. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

This paper investigates the global geometry of the group of Hamiltonian diffeomorphisms $\operatorname{Ham}(M,\omega)$ using random walks. On a large class of symplectic manifolds, we show that the expected Hofer norm of such a random walk grows at least as fast as the square root of the number of steps. Furthermore, we show that if the random walk is restricted to an abelian subgroup, then the growth rate is also bounded from above by the square root of the number of steps. We also provide some numerical evidence suggesting that this upper bound fails away from commutative subgroups. This provides, for subgroups of the group of Hamiltonian diffeomorphisms, a probabilistic version of the flatness observed in commutative finite-dimensional Lie groups. En route, we show that the class of probability measures introduced in the prequel is a class of Borel measures with respect to the $C^\infty$-topology on $\operatorname{Ham}(M,\omega)$, show the measurability of the stable commutator length, and show a central limit theorem for Hofer-Lipschitz quasimorphisms on $\operatorname{Ham}(M,\omega)$.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-08-11 in math.PR).
Categorie arXiv: Symplectic Geometry (math.SG) ; Dynamical Systems (math.DS); Probability (math.PR).
Pagina del lavoro: https://arxiv.org/abs/2608.07673 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-19T09:04:32.193Z dalla pagina https://arxiv.org/catchup/math.PR/2026-08-11?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).