共形不变度量在CLE₄上的研究 II：测地线的存在性
The conformally invariant metric on CLE$_4$ II: existence of geodesics
AI总结：
该系列第二篇论文证明了参数κ=4的共形环系综（CLE₄）中测地线的存在性，确立了其度量几何的定量估计，支撑了CLE₄唯一确定共形不变局部测地度量的结论。
AI中文摘要：
我们继续研究参数κ=4的共形环系综（CLE），这是一个临界阈值，在该阈值及以下时，环是简单且互不相交的，既不接触彼此也不接触区域边界。本文是三篇系列论文中的第二篇，旨在证明CLE₄的环唯一确定一种共形不变、局部且测地的度量，使得从区域边界出发的度量球增长与Werner和Wu的均匀探索一致。在这篇第二篇论文中，我们证明了测地线的存在性，表明任意两个环之间的任何测地线都支撑在CLE₄的环上（除了一个Hausdorff维数为零的集合），且不与区域边界相交。在此过程中，我们建立了CLE₄度量几何的精确定量估计，包括矩形距离的指数尾界以及度量球的多尺度四臂SLE₄非相交界。
英文摘要：
We continue our study of the conformal loop ensemble (CLE) with parameter $κ=4$, the critical threshold at or below which the loops are simple and disjoint, touching neither each other nor the domain boundary. This paper is the second in a series of three establishing that the loops of a CLE$_4$ uniquely determine a conformally invariant, local, and geodesic metric such that the metric ball growth from the domain boundary coincides with the uniform exploration of Werner and Wu. In this second paper, we prove the existence of geodesics, showing that any geodesic between two loops is supported on the CLE$_4$ loops (off a set of Hausdorff dimension zero) and does not intersect the domain boundary. Along the way, we establish sharp quantitative estimates for the CLE$_4$ metric geometry, including exponential tail bounds for rectangle distances and multi-scale four-arm SLE$_4$ non-intersection bounds for metric balls.
