论文标题
liouville量子重力通过保形回路合奏嵌套统计加权
Liouville quantum gravity weighted by conformal loop ensemble nesting statistics
论文作者
论文摘要
我们研究了Liouville量子重力（LQG）表面，其定律根据\ in \ Mathbb {n} _0 $标记点$ z_1，\ dots，z_n $在\ mathbb {n} _0 $标记点$ N} _0 $标记点$ n \的嵌套统计数据中已重量。这个想法是考虑$ \ prod_ {b \ subseteq \ {1，\ dots，n \}} e^{σ_bn_b} $，其中$σ_b\ in \ mathbb {r} $和$ n_b $是$ z__i的cle $ i的数量。这是通过近似程序精确的，其中作为证明的一部分，我们得出CLE的强空间独立性结果。重新加权诱导了liouville领域的对数奇异性，$ z_1，\ dots，z_n $，根据$σ_1，\ dots，σ_n$明确地具有幅度。我们定义表面的分区函数，计算\ {0,1 \} $中的$ n \，并根据低阶分区函数来表达$ n> 1 $点分区函数的递归公式。后一个结果的证明是基于Miller，Sheffield和Werner先前研究的连续剥离过程$ n = 0 $，我们为边界长度过程的生成器提供了一个明确的公式，该公式可以与一般$ n $的勘探相关联。我们使用递归公式进行部分表征$（σ_b\ colon B \ subseteq \ {1，\ dots，n \}）$的值。最后，我们为CLE的保形半径定律提供了新的证据，该法律最初是由Schramm，Sheffield和Wilson建立的。
We study Liouville quantum gravity (LQG) surfaces whose law has been reweighted according to nesting statistics for a conformal loop ensemble (CLE) relative to $n\in \mathbb{N}_0$ marked points $z_1,\dots,z_n$. The idea is to consider a reweighting by $\prod_{B\subseteq \{1,\dots,n\}} e^{σ_B N_B}$, where $σ_B\in\mathbb{R}$ and $N_B$ is the number of CLE loops surrounding the points $z_i$ for $i\in B$. This is made precise via an approximation procedure where as part of the proof we derive strong spatial independence results for CLE. The reweighting induces logarithmic singularities for the Liouville field at $z_1,\dots,z_n$ with a magnitude depending explicitly on $σ_1,\dots,σ_n$. We define the partition function of the surface, compute it for $n\in\{0,1\}$, and derive a recursive formula expressing the $n>1$ point partition function in terms of lower-order partition functions. The proof of the latter result is based on a continuum peeling process previously studied by Miller, Sheffield and Werner in the case $n=0$, and we derive an explicit formula for the generator of a boundary length process that can be associated with the exploration for general $n$. We use the recursive formula to partly characterize for which values of $(σ_B\colon B\subseteq \{1,\dots,n\})$ the partition function is finite. Finally, we give a new proof for the law of the conformal radius of CLE, which was originally established by Schramm, Sheffield, and Wilson.
