We study the structure of the Liouville quantum gravity (LQG) surfaces that are cut out as one explores a conformal loop-ensemble \(\hbox {CLE}_{\kappa '}\) for \(\kappa '\) in (4, 8) that is drawn on an independent \(\gamma \)-LQG surface for \(\gamma ^2=16/\kappa '\). The results are similar in flavor to the ones from our companion paper dealing with \(\hbox {CLE}_{\kappa }\) for \(\kappa \) 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 \(\hbox {CLE}_{\kappa '}\) 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 “CLE Percolations” described the law of interfaces obtained when coloring the loops of a \(\hbox {CLE}_{\kappa '}\) independently into two colors with respective probabilities p and \(1-p\). This description was complete up to one missing parameter \(\rho \). The results of the present paper about CLE on LQG allow us to determine its value in terms of p and \(\kappa '\). It shows in particular that \(\hbox {CLE}_{\kappa '}\) and \(\hbox {CLE}_{16/\kappa '}\) are related via a continuum analog of the Edwards-Sokal coupling between \(\hbox {FK}_q\) percolation and the q-state Potts model (which makes sense even for non-integer q between 1 and 4) if and only if \(q=4\cos ^2(4\pi / \kappa ')\). This provides further evidence for the long-standing belief that \(\hbox {CLE}_{\kappa '}\) and \(\hbox {CLE}_{16/\kappa '}\) represent the scaling limits of \(\hbox {FK}_q\) percolation and the q-Potts model when q and \(\kappa '\) are related in this way. Another consequence of the formula for \(\rho (p,\kappa ')\) 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.

我们研究了在( 4, 8) 为\(\gamma ^2=16/\kappa '\)绘制在独立的\(\gamma \) -LQG 表面上。结果与我们在 (8/3, 4) 中处理\(\hbox {CLE}_{\kappa }\) for \(\kappa \)的配套论文中的结果相似，其中CLE 是不相交且简单的。特别是，我们对 LQG 表面和\(\hbox {CLE}_{\kappa '}\)的组合结构进行编码就稳定生长-碎片树或其变体而言，它们也出现在装饰平面图上剥离过程的渐近研究中。这对先验不涉及 LQG 表面的问题产生了影响：在我们的题为“ CLE Percolations ”的论文中，描述了独立为\(\hbox {CLE}_{\kappa '}\)的循环着色时获得的界面定律分成具有各自概率p和\(1-p\) 的两种颜色。这个描述是完整的，直到缺少一个参数\(\rho \)。本文关于 LQG 上 CLE 的结果允许我们根据p和\(\kappa '\)来确定它的值。它特别表明\(\hbox {CLE}_{\kappa '}\)和\(\hbox {CLE}_{16/\kappa '}\)通过\(\hbox {FK}_q\)渗透和q状态 Potts 模型（即使对于 1 和 4 之间的非整数q也有意义）当且仅当\(q=4\cos ^2(4\pi / \kappa ' )\)。这为长期以来的信念提供了进一步的证据，即\(\hbox {CLE}_{\kappa '}\)和\(\hbox {CLE}_{16/\kappa '}\)代表\ (\hbox {FK}_q\)渗流和当q和\(\kappa '\)时的q -Potts 模型以这种方式相关。\(\rho (p,\kappa ')\)公式的另一个结果是这种分色模型（又名模糊 Potts 模型）的半平面臂指数值形式比二维模型的通常临界指数。