Conformal invariance of double random currents I: Identification of the limit

This is the first of two papers devoted to the proof of conformal invariance of the critical double random current model on the square lattice. More precisely, we show the convergence of loop ensembles obtained by taking the cluster boundaries in the sum of two independent currents with free and wired boundary conditions. The strategy is first to prove convergence of the associated height function to the continuum Gaussian free field, and then to characterise the scaling limit of the loop ensembles as certain local sets of this Gaussian free field. In this paper, we identify uniquely the possible subsequential limits of the loop ensembles. Combined with Duminil-Copin et al., this completes the proof of conformal invariance. © 2025 The Author(s). Proceedings of the London Mathematical Society is copyright © London Mathematical Society.