Double dimers, conformal loop ensembles and isomonodromic deformations

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. Kenyon established conformal invariance in the small mesh limit by considering topological observables of the model parameterized by $\SL_2(\C)$ representations of the fundamental group of the punctured domain. The scaling limit is conjectured to be $\CLE_4$, the Conformal Loop Ensemble at $\kappa=4$. 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 $\tau$-functions introduced by Jimbo, Miwa and Ueno in the context of isomonodromic deformations.