AI 中文总结
本文针对刘维尔量子引力模型，开发技术得到测地线内部段极值统计量的零一律，提升了Gwynne-Pfeffer-Sheffield 2022年的结果，所用技术由Bhatia-Kavvadias 2025年提出。
AI 中文摘要
在随机几何中，测地线常倾向于聚合在一起，且会穿越相对于典型环境高度奇异的区域。本文针对刘维尔量子引力模型，开发了一种技术，可得到沿测地线内部段测量的诸多极值统计量的零一律。具体而言，针对欧氏维数、相对于欧氏度量的最优赫尔德连续指数，以及基础高斯自由场（GFF）的最小/最大遭遇厚度等统计量，我们对测地线主体中的段得到了上述零一律，从而提升了Gwynne-Pfeffer-Sheffield在2022年的结果。所用的主要技术由Bhatia-Kavvadias在2025年开发，即沿长测地线的内部段布置大量小规模“典型”且行为良好的测地线。
英文摘要
In random geometry, geodesics often have a tendency to coalesce together and traverse regions highly singular relative to typical environments. In this paper, working with the model of Liouville quantum gravity, we develop a technique which yields zero-one laws for many extremal statistics measured along interior segments of the geodesic. Namely, working with statistics such as the Euclidean dimension, optimal Hölder continuity exponents with respect to the Euclidean metric and the minimal/maximal encountered thickness for the underlying GFF, we obtain zero-one laws for the above for segments in the bulk of geodesics, thereby upgrading the results of Gwynne-Pfeffer-Sheffield '22. The primary technique used, developed in Bhatia-Kavvadias '25, is to lay down a large family of small-scale "typical" and well-behaved geodesics along interior segments of a long geodesic.
Comments14 pages, 3 figures
