Tau-functions \`a la Dub\'edat and probabilities of cylindrical events for double-dimers and CLE(4)

Building upon recent results of Dub\'edat (see arXiv:1403.6076) 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 \emph{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 $\mathrm{Hom}(\pi_1(\Omega\setminus\{\lambda_1,\dots,\lambda_n\})\to\mathrm{SL}_2(\mathbb C))$ and on its (non-smooth) subvariety of locally unipotent representations. In particular, we do \emph{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 isomonodormic tau-function studied by Dub\'edat 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 $\Omega$ requires a small additional input, namely a mild crossing estimate for this nested conformal loop ensemble.