双二聚体和CLE（4）的Tau函数和圆柱事件的概率
Tau-functions à la Dubédat and probabilities of cylindrical events for double-dimers and CLE(4)
摘要：
基于Dubédat（见arXiv：1403.6076）关于Temperleyan近似$Ω^δ$到单连通域$Ω\subset\mathbb C$的双二聚体模型中拓扑算子收敛性的最新结果，我们证明了\emph{double-dimer loop ensembles}在$Ω^δ$上的圆柱事件概率收敛为$δ\to 0$。更确切地说，令$λ_1,\dots,λ_n\inΩ$和$L$为$Ω\setminus\{λ_1,\dots,λ_n\}$上的宏观层压，即，围绕至少两个被认为是同伦的穿刺的不相交的简单环的集合。我们发现，概率$P_L^δ$，一个获得$L$后，撤回所有的循环周围不超过一个穿刺从$Ω^δ$上的双二聚体循环合奏收敛到一个共形不变的限制$P_L$为$δ\to 0$，对于每个$L$。虽然我们的主要动机来自2D统计力学和概率，但证明是纯分析性质的。关键技术是对表示簇$\mathrm{Hom}(π_1(Ω\setminus\{λ_1,\dots,λ_n\})\to\mathrm{SL}_2(\mathbb C))$及其局部幂单表示的（非光滑）子簇上的整函数的分析。特别是，我们做\emph{not}使用任何双二聚体的RSW类型参数。极限$P_L$的概率$P_L^δ$被定义为系数的isomonodormic τ-功能研究Dubédat相对于福克-Goncharov层压基础上的代表品种。$P_L$与从$Ω$中嵌套CLE（4）的样本中获得$L$的概率相一致的事实需要少量的额外输入，即对该嵌套共形循环系综进行温和的交叉估计。
英文摘要：
Building upon recent results of Dubédat (see arXiv:1403.6076) on the convergence of topological correlators in the double-dimer model considered on Temperleyan approximations $Ω^δ$ to a simply connected domain $Ω\subset\mathbb C$ we prove the convergence of probabilities of cylindrical events for the \emph{double-dimer loop ensembles} on $Ω^δ$ as $δ\to 0$. More precisely, let $λ_1,\dots,λ_n\inΩ$ and $L$ be a macroscopic lamination on $Ω\setminus\{λ_1,\dots,λ_n\}$, i.e., a collection of disjoint simple loops surrounding at least two punctures considered up to homotopies. We show that the probabilities $P_L^δ$ that one obtains $L$ after withdrawing all loops surrounding no more than one puncture from a double-dimer loop ensemble on $Ω^δ$ converge to a conformally invariant limit $P_L$ as $δ\to 0$, for each $L$. Though our primary motivation comes from 2D statistical mechanics and probability, the proofs are of a purely analytic nature. The key techniques are the analysis of entire functions on the representation variety $\mathrm{Hom}(π_1(Ω\setminus\{λ_1,\dots,λ_n\})\to\mathrm{SL}_2(\mathbb C))$ and on its (non-smooth) subvariety of locally unipotent representations. In particular, we do \emph{not} use any RSW-type arguments for double-dimers. The limits $P_L$ of the probabilities $P_L^δ$ are defined as coefficients of the isomonodormic tau-function studied by Dubédat with respect to the Fock--Goncharov lamination basis on the representation variety. The fact that $P_L$ coincides with the probability to obtain $L$ from a sample of the nested CLE(4) in $Ω$ requires a small additional input, namely a mild crossing estimate for this nested conformal loop ensemble.
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