Paper detail

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

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.