Tau-functions à la Dubédat and probabilities of cylindrical events for double-dimers and CLE(4)

Building upon recent results of Dubédat [7] on the convergence topological correlators in double-dimer model considered Temperleyan approximations $\Omega^\delta$ to a simply connected domain $\Omega\subset\mathbb C$ we prove probabilities cylindrical events for loop ensembles as $\delta\to 0$. More precisely, let $\lambda_1,\dots,\lambda_n\in\Omega$ and $L$ be macroscopic lamination $\Omega\setminus\{\lambda_1,\dots,\lambda_n\}$, i.e., collection disjoint simple loops surrounding at least two punctures up homotopies. We show that $P_L^\delta$ one obtains after withdrawing all no more than puncture from ensemble converge conformally invariant limit $P_L$ as~$\delta \to 0$, each $L$.