Double dimers, conformal loop ensembles and isomonodromic deformations

Abstract

The double-dimer model consists in superimposing two independent, identically distributed perfect matchings on a planar graph, which produces an ensemble of non-intersecting loops. In [20], Kenyon established conformal invariance in the small mesh limit by considering topological observables of the model parameterized by SL${}_2(\mathbb{C})$ representations of the fundamental group of the punctured domain. The scaling limit is conjectured to be CLE${}_4$, the Conformal Loop Ensemble at $\kappa =4$ [36]. In support of this conjecture, we prove that a large subclass of these topological correlators converge to their putative CLE${}_4$ limit. Both the small mesh limit of the double-dimer correlators and the corresponding CLE${}_4$ correlators are identified in terms of the  -functions introduced by Jimbo, Miwa and Ueno [14] in the context of isomonodromic deformations.