  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.
