AI 中文总结
该研究在高维庞加莱球上分析欧几里得β型径向测度的极限行为，确定了四种弱锥极限情况，并给出收敛至直径至多为1的锥的尖锐准则。
AI 中文摘要
在高维庞加莱球上，欧几里得β型径向测度集中于一个球面，但双曲距离会放大剩余的径向扩散，因此常规的壳约化会丢失部分极限度量。在径向宽度与放大的角分离之间平衡的每个 regime（ regime 译为“ regime”，保留原词）中，我们按集中与耗散之间转变的阶确定重标空间的弱锥极限。四种可能性分别是有限星树生成的锥、直径至多为1的空间的锥、高斯锥的度量变换以及高斯锥。每棵星树具有从公共中心出发的分支、沿分支的移位指数分布，以及不同分支间通过中心的路径。我们还给出了收敛至直径至多为1的锥的尖锐准则。
英文摘要
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.
Comments41 pages, 1 figure
