CLE$_4$上的共形不变度量 III：唯一性
The conformally invariant metric on CLE$_4$ III: uniqueness
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中文总结 AI 辅助
本文是CLE₄共形不变度量系列论文的终篇，证明了前序构造的度量及其测地线的唯一性，确立重整化CLE_κ图度量在κ↓4时无需取子列即可收敛，给出了由均匀探测决定的测地线表示。
中文摘要 AI 辅助
本文是构建临界参数$κ=4$的共形环系综（conformal loop ensemble，CLE）环集上典范共形不变度量的系列论文的第三篇也是最后一篇。前两篇论文通过令$κ\downarrow4$时CLE$_κ$环集上重整化图度量的子序列极限，构造了CLE$_4$环集上的一种共形不变的局部度量，其从区域边界出发的度量球增长与Werner和Wu的均匀探测结果一致。在本文中，我们证明该度量及其测地线可由其性质唯一刻画，且它是CLE$_4$的可测函数。特别地，我们证明当$κ\downarrow4$时，重整化CLE$_κ$图度量无需取子序列即可收敛。证明的关键步骤是证明该度量由每个环到区域边界的测地线决定，而这些测地线又由均匀探测决定；这一表示将在未来工作中具有重要应用。
英文摘要
This paper is the third and final article in a series of papers constructing the canonical conformally invariant metric on the set of loops of the conformal loop ensemble (CLE) with critical parameter $κ=4$. The previous two articles construct, as a subsequential limit of the renormalized graph metric on the loops of CLE$_κ$ as $κ\downarrow 4$, a conformally invariant, local metric on the loops of a CLE$_4$ whose metric ball growth from the domain boundary coincides with the uniform exploration of Werner and Wu. In this paper, we establish that this metric is uniquely characterized by its properties, as are its geodesics, and that it is a measurable function of the CLE$_4$. In particular, we show that the renormalized CLE$_κ$ graph metric converges as $κ\downarrow 4$ without passing to a subsequence. A key step in the proof is to show that the metric is determined by the geodesics from each loop to the domain boundary, which are in turn determined by the uniform exploration; this representation will have important applications in future work.
