Conformal invariance of double random currents II: tightness and properties in the discrete

TL;DR

This paper proves the conformal invariance of the critical double random current model on the square lattice by analyzing loop ensembles and their convergence to the Gaussian free field, establishing key crossing properties.

Contribution

It demonstrates the convergence of loop ensembles and height functions in the double random current model, advancing understanding of conformal invariance in lattice models.

Findings

Convergence of loop ensembles to continuum limits.

Height functions converge to Gaussian free field.

Derived crossing properties for the discrete model.

Abstract

This is the second of two papers devoted to the proof of conformal invariance of the critical double random current on the square lattice. More precisely, we show convergence of loop ensembles obtained by taking the cluster boundaries in the sum of two independent critical currents (both for 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 derive crossing properties of the discrete model required to prove this characterization.