2018/09/03 by Mikhail Basok, Basok, Mikhail, Dmitry Chelkak +1 · 3 citations
Chemistry · Mathematics · Physics and Astronomy · #32A15 #34M56 #82B20 #Advanced Algebra and Geometry #FOS: Physical sciences #Mathematical Physics (math-ph) #Molecular spectroscopy and chirality #math-ph #math.MP #msc:32A15 #msc:34M56 #msc:82B20
paper · pdf · doi:10.48550/arxiv.1809.00690
minor update following referee's comments (42 pages, 5 figures)
openalex publication_date 2018/09/03 · arxiv created 2020/09/10 · arxiv updated 2020/09/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/31
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 Ωδ to a simply connected domain Ω⊂\mathbb C we prove the convergence of probabilities of cylindrical events for the 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 PLδ 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\)\toSL2(\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 PL of the probabilities PLδ 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 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.