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关于刘维尔量子引力上的非简单共形环系综与共形环系综渗流界面定律

Non-simple conformal loop ensembles on Liouville quantum gravity and the law of CLE percolation interfaces.

作者信息

Miller Jason, Sheffield Scott, Werner Wendelin

机构信息

Statslab, Center for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WB UK.

Department of Mathematics, MIT, 77 Massachusetts Avenue, Cambridge, MA 02139 USA.

出版信息

Probab Theory Relat Fields. 2021;181(1-3):669-710. doi: 10.1007/s00440-021-01070-4. Epub 2021 Jun 26.

DOI:10.1007/s00440-021-01070-4
PMID:34840373
原文链接:https://pmc.ncbi.nlm.nih.gov/articles/PMC8595195/
Abstract

We study the structure of the Liouville quantum gravity (LQG) surfaces that are cut out as one explores a conformal loop-ensemble for in (4, 8) that is drawn on an independent -LQG surface for . The results are similar in flavor to the ones from our companion paper dealing with for in (8/3, 4), where the loops of the CLE are disjoint and simple. In particular, we encode the combined structure of the LQG surface and the in terms of stable growth-fragmentation trees or their variants, which also appear in the asymptotic study of peeling processes on decorated planar maps. This has consequences for questions that do a priori not involve LQG surfaces: In our paper entitled "" described the law of interfaces obtained when coloring the loops of a independently into two colors with respective probabilities and . This description was complete up to one missing parameter . The results of the present paper about CLE on LQG allow us to determine its value in terms of and . It shows in particular that and are related via a continuum analog of the Edwards-Sokal coupling between percolation and the -state Potts model (which makes sense even for non-integer between 1 and 4) if and only if . This provides further evidence for the long-standing belief that and represent the scaling limits of percolation and the -Potts model when and are related in this way. Another consequence of the formula for is the value of half-plane arm exponents for such divide-and-color models (a.k.a. fuzzy Potts models) that turn out to take a somewhat different form than the usual critical exponents for two-dimensional models.

摘要

我们研究了在一个独立的(\gamma)-李维尔量子引力（LQG）曲面上绘制的(\text{CLE}{4,8})共形环系综时所切割出的LQG曲面的结构。其结果在性质上与我们关于(\text{CLE}{8/3,4})的配套论文中的结果相似，在该论文中，CLE的环是不相交且简单的。特别地，我们用稳定增长-分裂树或其变体来编码LQG曲面和(\text{CLE})的组合结构，这也出现在装饰平面图上剥离过程的渐近研究中。这对一些先验上不涉及LQG曲面的问题有影响：在我们题为“……”的论文中，描述了将(\text{CLE})的环分别以概率(p)和(1 - p)独立染成两种颜色时所得到的界面的规律。该描述在一个缺失参数(\cdots)的情况下是完整的。本文关于LQG上的(\text{CLE})的结果使我们能够根据(p)和(\gamma)确定其值。特别地，它表明当且仅当(\cdots)时，(p)和(\gamma)通过渗流与(q)-态Potts模型之间的爱德华兹 - 索卡尔耦合的连续类似物相关（这甚至对于1到4之间的非整数(q)也有意义）。这为长期以来的信念提供了进一步的证据，即当(p)和(\gamma)以这种方式相关时，它们分别代表渗流和(q)-Potts模型的标度极限。(\cdots)公式的另一个结果是这种划分和染色模型（又名模糊Potts模型）的半平面臂指数的值，其结果与二维模型的通常临界指数的形式有所不同。

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