The Ising magnetisation field and the Gaussian free field

Abstract

We construct a natural coupling between the continuum Gaussian free field (GFF) and the critical Ising magnetisation field (IMF) in a planar domain. In fact, we show that two independent IMFs with $+$ boundary conditions and two independent IMFs with free boundary conditions are a deterministic function of a single instance of the GFF together with a sequence of independent coin flips. This construction should be seen as an extension of the bosonisation phenomenon, and to the best of our knowledge its existence has not been predicted before. We arrive at our main result in the continuum by studying novel discrete structures. Our starting point is a coupling resembling the Edwards-Sokal coupling between the Ising model and the Fortuin-Kasteleyn random cluster model, though with role of the latter played by a different percolation model obtained from the double random current model. By taking a scaling limit of the coupling at criticality, we obtain a continuum Edwards-Sokal-like representation of the IMFs in terms of certain two-valued sets of the GFF introduced by Aru, Sep\'ulveda and Werner.