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Sharp Typical Distance and Exponential Small-Ball Bounds in One-Step-Cliff Nielsen Geometry — arXiv:2609.09642

Source: arXiv (en)
Original: https://arxiv.org/abs/2609.09642
License (as archived): CC0-1.0
Archived: 2026-10-16T09:51:10.835Z
SHA-256 of the archived text: 83353f7a2f392604e0a3d2a516d69ddc3923f43acf31e7fa3fa22ac16ab6ab5b
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Sharp Typical Distance and Exponential Small-Ball Bounds in One-Step-Cliff Nielsen Geometry — arXiv:2609.09642 — arXiv (en), https://arxiv.org/abs/2609.09642 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_296292f032be (Metraipolis archive dc_296292f032be, sha256 83353f7a2f392604e0a3d2a516d69ddc3923f43acf31e7fa3fa22ac16ab6ab5b)

BibTeX

@misc{dc296292f032be,
  title = {Sharp Typical Distance and Exponential Small-Ball Bounds in One-Step-Cliff Nielsen Geometry — arXiv:2609.09642},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2609.09642},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_296292f032be; sha256 83353f7a2f392604e0a3d2a516d69ddc3923f43acf31e7fa3fa22ac16ab6ab5b},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2609.09642. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

Let $D=2^n$ and equip $\operatorname{PU}(D)$ with the one-step-cliff Nielsen metric, with quadratic metric coefficients one in Pauli directions of weights one and two and $D^2$ in all higher weights. We prove that the distance from the identity of a Haar-random element, normalized by $D$, converges to $\pi/\sqrt3$ in probability and in $L^p$ for every $1\le p<\infty$. Quantitatively, it lies within $O(D^{-1/8}(\log D)^{1/2})$ of this limit outside a set of Haar measure at most $\exp\{-\Omega(D^{7/4}\log D)\}$. For each fixed $0<x<\pi/\sqrt3$, the ball of radius $xD$ has Haar measure $\exp\{-\Theta_x(D^2)\}$; for $x>\pi/\sqrt3$, its complement has measure at most $e^{-c_xD^2}$ for some $c_x>0$. As $x\uparrow\pi/\sqrt3$, the lower and upper logarithmic rates are both asymptotic to $(\pi^2/3-x^2)^2/(16\zeta(3))$. The small-ball upper bound follows from a comparison of Jacobi determinants, obtained by rescaling the linearized geodesic equations and applying Kato transport. Weyl integration reduces the remaining integral to an Abel-regularized logarithmic-energy estimate on the circle. A centered principal logarithm and concentration of the circular unitary ensemble eigenangle second moment give the distance upper bound.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-09-10 in math.PR).
Categorie arXiv: Probability (math.PR) ; Mathematical Physics (math-ph).
Pagina del lavoro: https://arxiv.org/abs/2609.09642 — 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 2026-10-16T08:44:35.277Z dalla pagina https://arxiv.org/catchup/math.PR/2026-09-10?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).