Tau-functions à la Dubédat and probabilities of cylindrical events for double-dimers and CLE(4)
Tau-functions à la Dubédat and probabilities of cylindrical events for double-dimers and CLE(4)
Résumé
Building upon recent results of Dub\'edat on the convergence of topological correlators in the double-dimer model considered on Temperleyan approximations Ωδ to a simply connected domain Ω⊂C we prove the convergence of probabilities of cylindrical events for the \emph{double-dimer loop ensembles} on Ωδ as δ→0. More precisely, let λ1,…,λn∈Ω and L be a macroscopic lamination on Ω∖{λ1,…,λn}, i.e., a collection of disjoint simple loops surrounding at least two punctures considered up to homotopies. We show that the probabilities PδL that one obtains L after withdrawing all loops surrounding no more than one puncture from a double-dimer loop ensemble on Ωδ converge to a conformally invariant limit PL 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,…,λn})→SL2(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 PL of the probabilities PδL 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 PL 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.