论文标题
关于liouville量子重力和CLE渗透界面定律的非简单共形环路
Non-simple conformal loop ensembles on Liouville quantum gravity and the law of CLE percolation interfaces
论文作者
论文摘要
我们研究了liouville量子重力（LQG）表面的结构，这些表面被切割出来，这些表面探索了$（4,8）$中的$κ'$（4,8）$中的$γ$ -LQG表面上的$κ'$ $κ'$ $ _ {κ'} $，用于$γ^2 = 16/κ$。结果的风味与我们的论文中有关CLE $_κ$的$κ$（8/3,4）$的味道类似，那里的CLE环路是脱节和简单的。特别是，我们在稳定的生长裂片树或其变体方面编码了LQG表面和CLE $ _ {κ'} $的组合结构，这些结构也出现在装饰平面图上的剥离过程的渐近研究中。 这对不涉及LQG表面的问题产生了后果：我们以前的论文“ CLE Percolations”描述了当将CLE $ _ {κ'} $的环上涂上两种颜色的界面定律，并以各自的概率$ p $和$ 1-P $独立地分为两种颜色。此描述最多可达一个缺少的参数$ρ$。有关LQG CLE的本文的结果使我们能够根据$ P $和$κ$确定其价值。它特别表明cle $ _ {κ'} $和cle $ _ {16/κ'} $是通过fk $ _q $ percolation和$ q $ $ q $ - 州模型之间的爱德华兹 - 卡（Edwards-sokal）耦合的连续类似物（即使是$ $ $ $ $ $ 1 $ $ $ 4也有意义） $ q = 4 \ cos^2（4π/κ'）$。这为长期以来的信念提供了进一步的证据，即CLE $ _ {κ'} $和CLE $ _ {16/κ'} $代表Fk $ _Q $ Percolation的缩放限制，而当$ q $ -potts型号当$ q $和$ q $ and $κ$时。该公式的$ρ（P，κ'）$的另一个结果是对于此类分裂和颜色型号（又称fuzzy Potts型号）的半平面臂指数的值，结果与二维模型的常规关键指数有所不同。
We study the structure of the Liouville quantum gravity (LQG) surfaces that are cut out as one explores a conformal loop-ensemble CLE$_{κ'}$ for $κ'$ in $(4,8)$ that is drawn on an independent $γ$-LQG surface for $γ^2=16/κ'$. The results are similar in flavor to the ones from our paper dealing with CLE$_κ$ for $κ$ 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 CLE$_{κ'}$ 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: Our previous paper "CLE percolations" described the law of interfaces obtained when coloring the loops of a CLE$_{κ'}$ independently into two colors with respective probabilities $p$ and $1-p$. This description was complete up to one missing parameter $ρ$. The results of the present paper about CLE on LQG allow us to determine its value in terms of $p$ and $κ'$. It shows in particular that CLE$_{κ'}$ and CLE$_{16/κ'}$ are related via a continuum analog of the Edwards-Sokal coupling between 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 $q=4\cos^2(4π/κ')$. This provides further evidence for the long-standing belief that CLE$_{κ'}$ and CLE$_{16/κ'}$ represent the scaling limits of FK$_q$ percolation and the $q$-Potts model when $q$ and $κ'$ are related in this way. Another consequence of the formula for $ρ(p,κ')$ 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.
