Sharp Typical Distance and Exponential Small-Ball Bounds in One-Step-Cliff Nielsen Geometry

Abstract: Let $D=2<sup>n$ 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 $D<sup>2$ 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 $L<sup>p$ for every $1\le p&lt;\infty$. Quantitatively, it lies within $O(D^{-1/8}(\log D)<sup>{1/2})$ of this limit outside a set of Haar measure at most $\exp{-Ω(D^{7/4}\log D)}$. For each fixed $0&lt;x&lt;π/\sqrt3$, the ball of radius $xD$ has Haar measure $\exp\{-Θ_x(D^2)\}$; for $x&gt;π/\sqrt3$, its complement has measure at most $e<sup>{-c_xD<sup>2}$ for some $c_x&gt;0$. As $x\uparrowπ/\sqrt3$, the lower and upper logarithmic rates are both asymptotic to $(π<sup>2/3-x<sup>2)<sup>2/(16ζ(3))$. 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.