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

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.

arXiv comment53 pages

References in corpus (5)

• Marginal triviality of the scaling limits of critical 4D Ising and ${\varphi }_4^4$ models
• Random Currents and Continuity of Ising Model's Spontaneous Magnetization
• A note on Ising random currents, Ising-FK, loop-soups and the Gaussian free field
• Planar random-cluster model: fractal properties of the critical phase
• The planar Ising model and total positivity