论文标题
简单的共形环与liouville量子重力
Simple Conformal Loop Ensembles on Liouville Quantum Gravity
论文作者
论文摘要
我们表明，当一个人在独立的$ \sqrtκ$ -liouville量子引力（LQG）上绘制简单的形状环（$κ\ for $κ\ for $κ\ for $κ\ in（8/3,4）$）时磁盘。这种结构使我们能够在LQG上的CLE，不对称$（4/κ）$稳定的过程和标记的分支树之间直接联系。这些过程的正和负跳跃强度之间的比率是$ - \ cos（4π/κ）$，可以解释为在LQG设置的CLE中的CLE环的“密度”。正跳跃对应于发现CLE循环的发现（循环的LQG长度由跳跃大小给出），而负跳跃对应于发现过程将其余的被发现的域分解为两个部分的矩。 一些后果是：（i）它为量子磁盘的拼布/焊接提供了CLE的结构。 （ii）它允许构建生活在CLE地毯上的“自然量子测量”。 （iii）它使我们能够为SLE过程和CLE本身（无LQG）得出一些新的特性和公式，例如一般不对称SLE $_κ（κ-6）$过程的确切分布。 目前的工作直接涉及连续体中的结构，也没有提及离散模型，但是我们的计算与在大面上的平面图上的O（n）模型的缩放限制相匹配，而LQG上的CLE和CLE。的确，我们的莱维树描述正是在研究大规模剥离的剥离限制中的分离装饰平面地图的研究中，例如在最近的Bertoin，Budd，Curien和Kortchemski的工作中。 在LQG上，非简单的cles的情况是另一篇论文的话题。
We show that when one draws a simple conformal loop ensemble (CLE$_κ$ for $κ\in (8/3,4)$) on an independent $\sqrtκ$-Liouville quantum gravity (LQG) surface and explores the CLE in a natural Markovian way, the quantum surfaces (e.g., corresponding to the interior of the CLE loops) that are cut out form a Poisson point process of quantum disks. This construction allows us to make direct links between CLE on LQG, asymmetric $(4/κ)$-stable processes, and labeled branching trees. The ratio between positive and negative jump intensities of these processes turns out to be $-\cos (4 π/ κ)$, which can be interpreted as a "density" of CLE loops in the CLE on LQG setting. Positive jumps correspond to the discovery of a CLE loop (where the LQG length of the loop is given by the jump size) and negative jumps correspond to the moments where the discovery process splits the remaining to be discovered domain into two pieces. Some consequences are the following: (i) It provides a construction of a CLE on LQG as a patchwork/welding of quantum disks. (ii) It allows to construct the "natural quantum measure" that lives in a CLE carpet. (iii) It enables us to derive some new properties and formulas for SLE processes and CLE themselves (without LQG) such as the exact distribution of the trunk of the general asymmetric SLE$_κ(κ-6)$ processes. The present work deals directly with structures in the continuum and makes no reference to discrete models, but our calculations match those for scaling limits of O(N) models on planar maps with large faces and CLE on LQG. Indeed, our Lévy-tree descriptions are exactly the ones that appear in the study of the large-scale limit of peeling of discrete decorated planar maps such as in recent work of Bertoin, Budd, Curien and Kortchemski. The case of non-simple CLEs on LQG is the topic of another paper.
