Inhomogeneous Gaussian Free Field inside the interacting arctic curve

Abstract

The six-vertex model with domain-wall boundary conditions is one representative of a class of two-dimensional lattice statistical mechanics models that exhibit a phase separation known as the arctic curve phenomenon. In the thermodynamic limit, the degrees of freedom are completely frozen in a region near the boundary, while they are critically fluctuating in a central region. The arctic curve is the phase boundary that separates those two regions. Critical fluctuations inside the arctic curve have been studied extensively, both in physics and in mathematics, in free models (i.e., models that map to free fermions, or equivalently to determinantal point processes). Here we study those critical fluctuations in the interacting (i.e., not free, not determinantal) six-vertex model, and provide evidence for the following two claims: (i) the critical fluctuations are given by a Gaussian Free Field (GFF), as in the free case, but (ii) contrarily to the free case, the GFF is inhomogeneous, meaning that its coupling constant $K$ becomes position-dependent, $K \rightarrow K({\rm x})$. The evidence is mainly based on the numerical solution of appropriate Bethe ansatz equations with an imaginary extensive twist, and on transfer matrix computations, but the second claim is also supported by the analytic calculation of $K$ and its first two derivatives in selected points. Contrarily to the usual GFF, this inhomogeneous GFF is not defined in terms of the Green's function of the Laplacian $\Delta = \nabla \cdot \nabla$ inside the critical domain, but instead, of the Green's function of a generalized Laplacian $\Delta = \nabla \cdot \frac{1}{K} \nabla$ parametrized by the function $K$. Surprisingly, we also find that there is a change of regime when $\Delta \leq -1/2$, with $K$ becoming singular at one point.