发表机构
Institute for Theoretical Sciences, Westlake University（西湖大学理论科学研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明单步悬崖Nielsen度量下Haar随机元素到恒等元的归一化距离依概率收敛于$\pi/\sqrt3$，并给出指数级小球的精确测度界，方法结合Jacobi行列式比较与Weyl积分。
AI 中文摘要
设$D=2^n$，并在$\operatorname{PU}(D)$上赋予单步悬崖Nielsen度量，其二次度量系数在权重一和二的Pauli方向上为1，在所有更高权重方向上为$D^2$。我们证明，Haar随机元素到恒等元的距离除以$D$后，依概率及对所有$1\le p<\infty$依$L^p$收敛到$\pi/\sqrt3$。定量地，在Haar测度至多为$\exp\{-\Omega(D^{7/4}\log D)\}$的集合之外，该距离位于此极限的$O(D^{-1/8}(\log D)^{1/2})$范围内。对每个固定的$0<x<\pi/\sqrt3$，半径为$xD$的球的Haar测度为$\exp\{-\Theta_x(D^2)\}$；对$x>\pi/\sqrt3$，其补集的测度至多为$e^{-c_xD^2}$，其中$c_x>0$。当$x\uparrow\pi/\sqrt3$时，上下对数速率均渐近于$(\pi^2/3-x^2)^2/(16\zeta(3))$。小球上界通过比较Jacobi行列式得到，该比较通过对线性化测地线方程进行重标度并应用Kato输运实现。Weyl积分将剩余积分化为圆上的Abel正则化对数能量估计。中心主对数及圆酉系综本征角二阶矩的集中性给出距离上界。
英文摘要
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 $π/\sqrt3$ 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 $\exp\{-Ω(D^{7/4}\log D)\}$. For each fixed $0<x<π/\sqrt3$, the ball of radius $xD$ has Haar measure $\exp\{-Θ_x(D^2)\}$; for $x>π/\sqrt3$, its complement has measure at most $e^{-c_xD^2}$ for some $c_x>0$. As $x\uparrowπ/\sqrt3$, 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.
Comments38 pages, no figures
