Poincaré Beta Balls: Radial Laws, Shell Transforms, and Phase Diagram

Abstract

On a high-dimensional Poincaré ball, a Euclidean beta-type radial measure concentrates near a sphere, but hyperbolic distance amplifies the surviving radial spread, so the usual shell reduction loses part of the limiting metric. In each regime of the balance between this radial width and amplified angular separation, we determine the weak pyramid limit of the spaces rescaled at the order at which the transition between concentration and dissipation occurs. The four possibilities are a pyramid generated by finite star trees, the pyramid of spaces of diameter at most one, metric transforms of the Gaussian pyramid, and the Gaussian pyramid. Each star tree has branches from a common center, a shifted exponential distribution along them, and paths between different branches through the center. We also give a sharp criterion for convergence to the diameter-at-most-one pyramid.

Disclosure

“assertion. This completes the proof. □ Acknowledgments The author would like to thank Professor Takashi Shioya for many helpful sug- gestions and guidance. The author used Claude, GPT-5.5, and GPT-5.6-series Codex models as AI-assisted tools in preparing this manuscript. The author re- viewed and revised the mathematical content and takes full responsibility for the final manuscript.”

PDF page 40

Classification

Rewriting existing author-written text

Multiplier

4

Verified

2026-08-23T19:12:37.971113Z

Structural counts