Brownian half-plane excursion and critical Liouville quantum gravity
Brownian half-plane excursion and critical Liouville quantum gravity
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布朗半平面偏移和临界刘维尔量子引力
DOI：
10.1112/jlms.12689
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发表时间：
2022
期刊：
影响因子：
--
通讯作者：
Aru J
中科院分区：
文献类型：
--
作者：
Aru J
在一项开创性的工作中，杜普兰蒂尔、米勒和谢菲尔德证明，亚临界刘维尔量子引力 (LQG) 与施拉姆-洛纳演化 (SLE) 的结合可以通过将一对布朗运动粘合在一起来获得。在本文中，我们通过限制性论证研究了他们在关键案例中的对应结果。特别是，我们证明，当在亚临界设置中发送 κ′↓4$\kappa ^{\prime } \downarrow 4$ 时，磁盘中的空间填充 SLEκ′$_{\kappa ^{\prime }}$ 退化为 Werner 和 Wu 引入的 CLE4$_4$（其中 CLE 是共形环系综）探索，以及一组独立且相同分布的硬币抛掷按探索的分支点进行索引。此外，在相同的极限下，我们观察到，虽然一对初始布朗运动折叠成一个，但仍然可以从这对中提取两个不同的独立布朗运动 (A,B)$(A,B)$，这样布朗运动 A$A$ 编码从 CLE 循环到圆盘边界的 LQG 距离，布朗运动 B$B$ 编码 CLE4$_4$ 循环的边界长度。与亚临界设置相反，(A,B)$(A,B)$ 对不能确定 CLE 修饰的 LQG 表面。我们的论文还讨论了与随机平面映射、保形不变 CLE4$_4$ 度量和增长碎片之间的关系。
In a groundbreaking work, Duplantier, Miller and Sheffield showed that subcritical Liouville quantum gravity (LQG) coupled with Schramm–Loewner evolutions (SLE) can be obtained by gluing together a pair of Brownian motions. In this paper, we study the counterpart of their result in the critical case via a limiting argument. In particular, we prove that as one sends κ′↓4$\kappa ^{\prime } \downarrow 4$ in the subcritical setting, the space‐filling SLEκ′$_{\kappa ^{\prime }}$ in a disk degenerates to the CLE4$_4$ (where CLE is conformal loop ensembles) exploration introduced by Werner and Wu, along with a collection of independent and identically distributed coin tosses indexed by the branch points of the exploration. Furthermore, in the same limit, we observe that although the pair of initial Brownian motions collapses to a single one, one can still extract two different independent Brownian motions (A,B)$(A,B)$ from this pair, such that the Brownian motion A$A$ encodes the LQG distance from the CLE loops to the boundary of the disk and the Brownian motion B$B$ encodes the boundary lengths of the CLE4$_4$ loops. In contrast to the subcritical setting, the pair (A,B)$(A,B)$ does not determine the CLE‐decorated LQG surface. Our paper also contains a discussion of relationships to random planar maps, the conformally invariant CLE4$_4$ metric and growth fragmentations.