2018/09/03 by Mikhail Basok, Basok, Mikhail, Dmitry Chelkak +1 · 1 citation
Chemistry · Mathematics · #32A15 #34M56 #82B20 #Advanced Algebra and Geometry #FOS: Physical sciences #Mathematical Physics (math-ph) #Molecular spectroscopy and chirality
paper · pdf · doi:10.48550/arxiv.1809.00690
openalex publication_date 2018/09/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/31
Building upon recent results of Dub 'edat (see arXiv:1403.6076) on the\nconvergence of topological correlators in the double-dimer model considered on\nTemperleyan approximations \Ω^\δ to a simply connected domain\n\Ω\⊂ mathbb C we prove the convergence of probabilities of\ncylindrical events for the \double-dimer loop ensembles on\n\Ω^\δ as \δ\→ 0. More precisely, let\n\λ1,\…,\λn\∈\Ω and L be a macroscopic lamination on\n\Ω\∖ \λ1,\…,\λn , i.e., a collection of disjoint\nsimple loops surrounding at least two punctures considered up to homotopies. We\nshow that the probabilities PL^\δ that one obtains L after withdrawing\nall loops surrounding no more than one puncture from a double-dimer loop\nensemble on \Ω^\δ converge to a conformally invariant limit PL as\n\δ \→ 0, for each L.\n Though our primary motivation comes from 2D statistical mechanics and\nprobability, the proofs are of a purely analytic nature. The key techniques are\nthe analysis of entire functions on the representation variety\n\Hom(\π1(\Ω\∖ \λ1,\…,\λn )\→\SL2( mathbb\nC)) and on its (non-smooth) subvariety of locally unipotent representations.\nIn particular, we do \not use any RSW-type arguments for double-dimers.\n The limits PL of the probabilities PL^\δ are defined as\ncoefficients of the isomonodormic tau-function studied by Dub 'edat with\nrespect to the Fock--Goncharov lamination basis on the representation variety.\nThe fact that PL coincides with the probability to obtain L from a sample\nof the nested CLE(4) in \Ω requires a small additional input, namely a\nmild crossing estimate for this nested conformal loop ensemble.\n