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

TL;DR

This paper proves the convergence of loop ensembles derived from double random currents on a square lattice to a conformally invariant limit, using Gaussian free fields and boundary conditions, as part of establishing conformal invariance.

Contribution

It uniquely identifies the subsequential limits of loop ensembles in the double random current model, advancing the proof of conformal invariance in this setting.

Findings

Convergence of height functions to the Gaussian free field.

Characterization of loop ensemble limits as local sets of the GFF.

Unique identification of subsequential limits of loop ensembles.

Abstract

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 characterize 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 the second paper, this completes the proof of conformal invariance.