Fluctuation scale of the true Haar-typical distance

Determine the fluctuation scale of the true distance from the identity to a Haar-random element in the one-step-cliff Nielsen metric, beyond the established concentration window of order D^{7/8}(\log D)^{1/2}.

Background

The paper proves that, for the one-step-cliff Nielsen metric on PU(D), the distance from the identity to a Haar-random projective unitary is concentrated around (\pi/\sqrt{3})D, with the currently established error window of order D{7/8}(\log D){1/2}.

The centered principal-logarithm path is shown to be asymptotically length minimizing to first order, and its length has order-one Gaussian fluctuations. However, the paper does not determine whether the true geodesic distance fluctuates on the same scale, nor does it identify the fluctuation law or a sharper concentration window. The authors therefore leave the fluctuation scale of the actual distance unresolved.

References

Determining the fluctuation scale of the true distance inside the present $D{7/8}(\log D){1/2}$ window remains open.

— Sharp Typical Distance and Exponential Small-Ball Bounds in One-Step-Cliff Nielsen Geometry  (2609.09642 - Fang, 9 Sep 2026) in Section 7, subsection “Further questions” (Section \ref{sec:circuit-outlook})