AI 中文总结
本文研究κ=4的共形环系综CLE₄，通过证明κ趋近于4⁺时CLE_κ环上图度量的重整化子序列极限存在，构造出CLE₄上共形不变的局部测地度量，其边界度量球增长与Werner和Wu的均匀探测一致。
AI 中文摘要
我们研究参数为$κ=4$的共形环系综（conformal loop ensemble，简称CLE），$κ=4$是临界值，当κ小于等于该值时，环是简单的，且互不相交，也不与区域边界相交。我们证明CLE$_4$的环可以唯一确定一个共形不变的局部测地度量，使得从区域边界出发的度量球增长与Werner和Wu的均匀探测过程一致。该度量此前由Sheffield、Watson和Wu在未发表工作中构造。我们的方法不同之处在于，我们证明该度量是当$κ\to 4^+$时，CLE$_κ$环上图度量经重整化后的极限。在这篇由三部分组成的系列论文的第一篇中，我们证明了子序列极限存在，并在CLE$_4$上定义了一个非平凡的共形不变度量，该度量具有局部性，且从边界出发的度量球增长由Werner和Wu的均匀探测过程给出。在后续工作中，我们将证明该子序列极限是真正的极限。
英文摘要
We consider the conformal loop ensemble (CLE) with the parameter $κ=4$, the critical value at or below which the loops are simple and do not intersect each other or the domain boundary. We show that the loops of a CLE$_4$ uniquely determine a conformally invariant, local, and geodesic metric so that the metric ball growth from the domain boundary coincides with the uniform exploration of Werner and Wu. This metric was previously constructed in unpublished work of Sheffield, Watson, and Wu. Our approach differs in that we show that the metric arises as the renormalized limit of the graph metric on CLE$_κ$ loops as $κ\downarrow 4$. In this first paper in a series of three, we prove that the subsequential limits exist and define a non-trivial conformally invariant metric on CLE$_4$ which is local and such that the metric ball growth from the boundary is given by the uniform exploration of Werner and Wu. In subsequent work, we will show that the subsequential limit exists as a true limit.
Comments92 pages, 10 figures
