Conformal loop ensembles (CLEs) are random collections of loops in a simply connected domain, whose laws are characterized by a natural conformal invariance property. The set of points not surrounded by any loop is a canonical random connected fractal set — a random and conformally invariant analog of the Sierpinski carpet or gasket.In the present paper, we derive a direct relationship between the CLEs with simple loops ($\text{CLE}_{\unicode[STIX]{x1D705}}$ for $\unicode[STIX]{x1D705}\in (8/3,4)$, whose loops are Schramm’s $\text{SLE}_{\unicode[STIX]{x1D705}}$-type curves) and the corresponding CLEs with nonsimple loops ($\text{CLE}_{\unicode[STIX]{x1D705}^{\prime }}$ with $\unicode[STIX]{x1D705}^{\prime }:=16/\unicode[STIX]{x1D705}\in (4,6)$, whose loops are $\text{SLE}_{\unicode[STIX]{x1D705}^{\prime }}$-type curves). This correspondence is the continuum analog of the Edwards–Sokal coupling between the $q$-state Potts model and the associated FK random cluster model, and its generalization to noninteger $q$.Like its discrete analog, our continuum correspondence has two directions. First, we show that for each $\unicode[STIX]{x1D705}\in (8/3,4)$, one can construct a variant of $\text{CLE}_{\unicode[STIX]{x1D705}}$ as follows: start with an instance of $\text{CLE}_{\unicode[STIX]{x1D705}^{\prime }}$, then use a biased coin to independently color each $\text{CLE}_{\unicode[STIX]{x1D705}^{\prime }}$ loop in one of two colors, and then consider the outer boundaries of the clusters of loops of a given color. Second, we show how to interpret $\text{CLE}_{\unicode[STIX]{x1D705}^{\prime }}$ loops as interfaces of a continuum analog of critical Bernoulli percolation within $\text{CLE}_{\unicode[STIX]{x1D705}}$ carpets — this is the first construction of continuum percolation on a fractal planar domain. It extends and generalizes the continuum percolation on open domains defined by $\text{SLE}_{6}$ and $\text{CLE}_{6}$.These constructions allow us to prove several conjectures made by the second author and provide new and perhaps surprising interpretations of the relationship between CLEs and the Gaussian free field. Along the way, we obtain new results about generalized $\text{SLE}_{\unicode[STIX]{x1D705}}(\unicode[STIX]{x1D70C})$ curves for $\unicode[STIX]{x1D70C}<-2$, such as their decomposition into collections of $\text{SLE}_{\unicode[STIX]{x1D705}}$-type ‘loops’ hanging off of $\text{SLE}_{\unicode[STIX]{x1D705}^{\prime }}$-type ‘trunks’, and vice versa (exchanging $\unicode[STIX]{x1D705}$ and $\unicode[STIX]{x1D705}^{\prime }$). We also define a continuous family of natural $\text{CLE}$ variants called boundary conformal loop ensembles (BCLEs) that share some (but not all) of the conformal symmetries that characterize $\text{CLE}$s, and that should be scaling limits of critical models with special boundary conditions. We extend the $\text{CLE}_{\unicode[STIX]{x1D705}}$/$\text{CLE}_{\unicode[STIX]{x1D705}^{\prime }}$ correspondence to a $\text{BCLE}_{\unicode[STIX]{x1D705}}$/$\text{BCLE}_{\unicode[STIX]{x1D705}^{\prime }}$ correspondence that makes sense for the wider range $\unicode[STIX]{x1D705}\in (2,4]$ and $\unicode[STIX]{x1D705}^{\prime }\in [4,8)$.

中文翻译：

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CLE 渗滤液

Conformal loop ensembles (CLE) 是简单连接域中循环的随机集合，其规律的特征在于自然的共形不变性。未被任何环包围的点集是规范的随机连通分形集——Sierpinski 地毯或垫圈的随机且保形不变的模拟。在本文中，我们推导出具有简单环的 CLE 之间的直接关系 ($\text{CLE}_{\unicode[STIX]{x1D705}}$为了$\unicode[STIX]{x1D705}\in (8/3,4)$，其循环是施拉姆的$\text{SLE}_{\unicode[STIX]{x1D705}}$-型曲线）和相应的具有非简单循环的 CLE（$\text{CLE}_{\unicode[STIX]{x1D705}^{\prime }}$和$\unicode[STIX]{x1D705}^{\prime }:=16/\unicode[STIX]{x1D705}\in (4,6)$，其循环是$\text{SLE}_{\unicode[STIX]{x1D705}^{\prime }}$型曲线）。这种对应是 Edwards-Sokal 耦合的连续模拟$q$-state Potts 模型和相关的 FK 随机聚类模型，及其对非整数的推广$q$.就像它的离散模拟一样，我们的连续对应有两个方向。首先，我们证明对于每个$\unicode[STIX]{x1D705}\in (8/3,4)$, 可以构造一个变体$\text{CLE}_{\unicode[STIX]{x1D705}}$如下：从一个实例开始$\text{CLE}_{\unicode[STIX]{x1D705}^{\prime }}$，然后使用有偏差的硬币来独立地给每一个上色$\text{CLE}_{\unicode[STIX]{x1D705}^{\prime }}$以两种颜色之一循环，然后考虑给定颜色的循环簇的外部边界。其次，我们展示如何解释$\text{CLE}_{\unicode[STIX]{x1D705}^{\prime }}$循环作为临界伯努利渗透的连续模拟的接口$\text{CLE}_{\unicode[STIX]{x1D705}}$地毯——这是第一次在地毯上构建连续渗透分形平面域。它扩展和概括了由定义的开放域上的连续渗透$\text{SLE}_{6}$和$\text{CLE}_{6}$.这些结构使我们能够证明第二作者提出的几个猜想，并为 CLE 和高斯自由场之间的关系提供新的可能令人惊讶的解释。一路走来，我们获得了关于泛化的新结果$\text{SLE}_{\unicode[STIX]{x1D705}}(\unicode[STIX]{x1D70C})$曲线为$\unicode[STIX]{x1D70C}<-2$，例如将它们分解为$\text{SLE}_{\unicode[STIX]{x1D705}}$-type 'loops' 挂在$\text{SLE}_{\unicode[STIX]{x1D705}^{\prime }}$-类型“树干”，和反之亦然（交换$\unicode[STIX]{x1D705}$和$\unicode[STIX]{x1D705}^{\prime }$）。我们还定义了一个连续的自然族$\文本{CLE}$称为边界共形环集合 (BCLE) 的变体共享一些（但不是全部）表征的共形对称性$\文本{CLE}$s，这应该是具有特殊边界条件的关键模型的缩放限制。我们扩展$\text{CLE}_{\unicode[STIX]{x1D705}}$/$\text{CLE}_{\unicode[STIX]{x1D705}^{\prime }}$对应一个$\text{BCLE}_{\unicode[STIX]{x1D705}}$/$\text{BCLE}_{\unicode[STIX]{x1D705}^{\prime }}$对更广泛的范围有意义的对应$\unicode[STIX]{x1D705}\in (2,4]$和$\unicode[STIX]{x1D705}^{\prime }\in [4,8)$.