The (XOR-)Ising model and the Gaussian free field

Abstract: We introduce a probabilistic coupling of several fundamental models in two-dimensional statistical mechanics: a Gaussian free field, four critical Ising fields, and two critical XOR-Ising fields. Such a coupling can be seen as an extension of the bosonization framework of the Ising model and heavily features the theory of level sets of the Gaussian free field.

We first present the discrete counterpart of the coupling, focusing on a new percolation-based representation of the Ising model related to double random currents. We then explain how to take the joint scaling limit of all fields, passing all the key properties of the coupling to the continuum. If time allows, we discuss generalizations of the coupling to the Ashkin-Teller model and/or the near-critical case.

Based on joint works with Lorca Heeney, Marcin Lis and Avelio Sepúlveda.