Article Dans Une Revue Journal of the European Mathematical Society Année : 2021

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 ΩC we prove the convergence of probabilities of cylindrical events for the \emph{double-dimer loop ensembles} on Ωδ as δ0. More precisely, let λ1,,λnΩ 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 δ0, 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.

Dates et versions

hal-02936267 , version 1 (11-09-2020)

Identifiants

Citer

Mikhail Basok, Dmitry Chelkak. Tau-functions à la Dubédat and probabilities of cylindrical events for double-dimers and CLE(4). Journal of the European Mathematical Society, 2021, 23 (8), pp.2787-2832. ⟨10.4171/JEMS/1072⟩. ⟨hal-02936267⟩
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