Excursion decomposition of the XOR-Ising model

Abstract

We study the excursion decomposition of the two-dimensional critical XOR-Ising model with either $+$ or free boundary conditions. In the first part, we construct the decomposition directly in the continuum. This construction relies on the identification of the XOR-Ising field with the cosine or sine of a Gaussian free field (GFF) $ϕ$ multiplied by $α= 1/\sqrt{2}$, and is obtained by an appropriate exploration of two-valued level sets of the GFF. More generally, the same construction applies to the fields $:\! \cos(αϕ) \!:$ and $:\! \sin(αϕ) \!:$ for any $α\in (0,1)$. In the second part, we show that the continuum excursion decomposition arises as the scaling limit of the double random current decomposition of the critical XOR-Ising model on the square lattice. To this end, we exploit the rich Markovian structure of the discrete decomposition and strengthen the convergence of the double random current height function to the continuum GFF by establishing joint convergence with its cosine and sine. We conjecture that for $α\in [1/2,\sqrt{3}/2)$ the continuum excursion decompositions arise as the scaling limit of those of the Ashkin-Teller polarisation field along its critical line.