Building upon recent results of Dubédat [7] on the convergence of topological correlators in the double-dimer model considered on Temperleyan approximations $\Omega^\delta$ to a simply connected domain $\Omega\subset\mathbb C$ we prove the convergence of probabilities of cylindrical events for the double-dimer loop ensembles on $\Omega^\delta$ as $\delta\to 0$. More precisely, let $\lambda_1,\dots,\lambda_n\in\Omega$ and $L$ be a macroscopic lamination on $\Omega\setminus\{\lambda_1,\dots,\lambda_n\}$, i.e., a collection of disjoint simple loops surrounding at least two punctures considered up to homotopies. We show that the probabilities $P_L^\delta$ that one obtains $L$ after withdrawing all loops surrounding no more than one puncture from a double-dimer loop ensemble on $\Omega^\delta$ converge to a conformally invariant limit $P_L$ as~$\delta \to 0$, 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 Hom$(\pi_1(\Omega\setminus\{\lambda_1,\dots,\lambda_n\})\to\mathrm{SL}_2(\mathbb C)$ and on its (non-smooth) subvariety of locally unipotent representations. In particular, we do not use any RSW-type arguments for double-dimers.

The limits $P_L$ of the probabilities $P_L^\delta$ are defined as coefficients of the isomonodromic tau-function studied in [7] with respect to the Fock–Goncharov lamination basis on the representation variety. The fact that $P_L$ coincides with the probability of obtaining $L$ from a sample of the nested CLE(4) in $\Omega$ requires a small additional input, namely a mild crossing estimate for this nested conformal loop ensemble.

基于 Dubédat [7] 在双二聚体模型中拓扑相关器收敛的最新结果，基于 Temperleyan 近似值 $\Omega^\delta$ 到简单连接域 $\Omega\subset\mathbb C$ 我们证明了收敛双二聚体环系综圆柱事件的概率在 $\Omega^\delta$ 上为 $\delta\to 0$。更准确地说，令 $\lambda_1,\dots,\lambda_n\in\Omega$ 和 $L$ 是 $\Omega\setminus\{\lambda_1,\dots,\lambda_n\}$ 上的宏观叠层，即一个集合围绕至少两个被认为是同伦的穿孔的不相交的简单循环。我们表明，在从 $\Omega^\delta$ 上的双二聚体循环系综中撤回围绕不超过一个穿孔的所有循环后获得 $L$ 的概率 $P_L^\delta$ 收敛到共形不变极限 $P_L $ as~$\delta \to 0$，对于每个$L$。

虽然我们的主要动机来自二维统计力学和概率，但证明是纯粹的分析性质。关键技术是对表示多样性 Hom$(\pi_1(\Omega\setminus\{\lambda_1,\dots,\lambda_n\})\to\mathrm{SL}_2(\mathbb C)$以及它的（非平滑）局部单能表示子变体。特别是，我们不使用任何 RSW 类型的参数来表示双二聚体。

概率 $P_L^\delta$ 的极限 $P_L$ 被定义为 [7] 中研究的等单 tau 函数的系数，该系数相对于 Fock-Goncharov 分层基础上的表示种类。$P_L$ 与从 $\Omega$ 中的嵌套 CLE(4) 的样本中获得 $L$ 的概率相吻合的事实需要一个小的额外输入，即这个嵌套保形环系综的温和交叉估计。