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

Abstract

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.