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

Building upon recent results of Dubedat [7] on the convergence of topological correlators in the double-dimer model considered on Temperleyan approximations Ω^δ to a simply connected domain Ω ⊂ ℂ we prove the convergence of probabilities of cylindrical events for the double-dimer loop ensembles on Ω^δ as δ → 0. More precisely, let λ₁,..., λ_n ∈ Ω and L be a macroscopic lamination on Ω \ {λ₁,..., 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 → 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(π₁(Ω \ {λ₁,..., λ_n}) → SL₂(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^δ 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 PL coincides with the probability of obtaining 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. [ABSTRACT FROM AUTHOR]