Poincaré Beta Balls: Radial Laws, Hyperbolic Shell Metric Transforms, and Critical-Limit Regimes

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 critical-limit regimes are the shifted-exponential railway pyramid, the diameter-at-most-one regime, the nonlinear Gaussian-transform regime, and the Gaussian regime. Each finite-branch railway mm-space generating the first pyramid 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. The resulting classification identifies the precise critical scales and clarifies when angular and radial fluctuations contribute simultaneously to the limiting geometry.

arXiv comment43 pages, 1 figure