径向双曲测度：壳几何、金字塔极限与高斯相变
Radial Hyperbolic Measures: Shell Geometry, Pyramid Limits, and Gaussian Phase Transitions
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中文总结 AI 辅助
该研究探讨高维双曲空间中径向测度的壳几何、金字塔极限与高斯相变，明确有效半径的作用，揭示两类高斯的临界阶差异及对应收敛性质。
中文摘要 AI 辅助
在高维双曲空间中，径向测度向壳的集中并不一定决定金字塔极限：角集中与双曲扩张会改变分离性。有效半径是正质量集实际发生分离的尺度，而原始半径未必能体现这一点。在消失的重标度与径向波动下，半径收敛会给出所有带直径界的度量测度空间向金字塔的弱收敛。在模态径向吉布斯壳处，固有壳曲率的指数衰减及维度归一化的切向Bakry-Émery Ricci曲率可恢复半径。源于双曲体积的高斯与通过包裹欧氏高斯得到的高斯具有不同的临界阶，在临界阶以下它们是Lévy的，在临界阶以上无限耗散，且在临界点收敛到相应的带直径界的金字塔。
英文摘要
In high-dimensional hyperbolic space, concentration of a radial measure near a shell need not determine the pyramid limit: angular concentration and hyperbolic expansion alter separation. An effective radius is the scale on which positive-mass sets actually separate, which the raw radius need not give. With vanishing rescaling and radial fluctuations, radius convergence gives weak convergence to the pyramid of all metric measure spaces with the resulting diameter bound. At modal radial Gibbs shells, exponential decay of intrinsic shell curvature and dimension-normalized tangential Bakry-Émery Ricci curvature recovers the radius. The Gaussian intrinsic to hyperbolic volume and that obtained by wrapping a Euclidean Gaussian have different critical orders. They are Lévy below those orders, infinitely dissipate above them, and at criticality converge to the corresponding diameter-bounded pyramids.
