Sharp Typical Distance and Exponential Small-Ball Bounds in One-Step-Cliff Nielsen Geometry
Abstract
Let D=2n and equip PU(D) with the one-step-cliff Nielsen metric, with quadratic metric coefficients one in Pauli directions of weights one and two and D2 in all higher weights. We prove that the distance from the identity of a Haar-random element, normalized by D, converges to π/√3 in probability and in Lp for every 1≤ p<∞. Quantitatively, it lies within O(D−1/8(log D)1/2) of this limit outside a set of Haar measure at most exp{−Ω(D7/4log D)}. For each fixed 0<x<π/√3, the ball of radius xD has Haar measure exp{−Θx(D2)}; for x>π/√3, its complement has measure at most e−cxD2 for some cx>0. As x↑π/√3, the lower and upper logarithmic rates are both asymptotic to (π2/3−x2)2/(16ζ(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.
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BibTeX
@misc{fang2026sharp,
title = {{Sharp Typical Distance and Exponential Small-Ball Bounds in One-Step-Cliff Nielsen Geometry}},
author = {Honghuai Fang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09642},
eprint = {2609.09642},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09642},
}
RIS
TY - UNPB
TI - Sharp Typical Distance and Exponential Small-Ball Bounds in One-Step-Cliff Nielsen Geometry
AU - Honghuai Fang
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.09642
UR - https://arxiv.org/abs/2609.09642
AB - Let D=2^n and equip 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 π/√3 in probability and in L^p for every 1≤ p<∞. Quantitatively, it lies within O(D^−1/8(log D)^1/2) of this limit outside a set of Haar measure at most exp{−Ω(D^7/4log D)}. For each fixed 0<x<π/√3, the ball of radius xD has Haar measure exp{−Θ_x(D^2)}; for x>π/√3, its complement has measure at most e^−c_xD^2 for some c_x>0. As x↑π/√3, the lower and upper logarithmic rates are both asymptotic to (π^2/3−x^2)^2/(16ζ(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.
ER -