Arctic curves of the four-vertex model

We consider the four-vertex model with a special choice of fixed boundary conditions giving rise to limit shape phenomena. More generally, the considered boundary conditions relate vertex models to scalar products of off-shell Bethe states, boxed plane partitions, and fishnet diagrams in quantum field theory. In the scaling limit, the model exhibits the emergence of an arctic curve separating a central disordered region from six frozen ‘corners’ of ferroelectric or anti-ferroelectric type. We determine the analytic expression of the interface by means of the Tangent Method. We supplement this heuristic method with an alternative, rigorous derivation of the arctic curve. This is based on the exact evaluation of suitable correlation functions, devised to detect spatial transition from order to disorder, in terms of the partition function of some discrete log-gas associated to the orthogonalizing measure of the Hahn polynomials. As a by-product, we also deduce that the arctic curve’s fluctuations are governed by the Tracy–Widom distribution.