Arctic curves of the four-vertex model

Abstract

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.

Limit shapes from harmonicity: dominos and the five vertex model

“…Its arctic curves for specific (analogous to boxed plane partition) boundary conditions were recently studied in [6]. In order to be consistent with [6] we also switch to north-and east-going lattice paths in this discussion, i.e. mirror the last two configurations with respect to the vertical axis in figure 10.…”

“…In this section, we consider the four-vertex model which is a degenerate case of the five-vertex model, where the third type of vertex in figure 10 has weight 0. Its arctic curves for specific (analogous to boxed plane partition) boundary conditions were recently studied in [6]. In order to be consistent with [6] we also switch to north-and east-going lattice paths in this discussion, i.e.…”

“…In fact, we observe that the surface tension σ has the same structure as the general large r five-vertex model (see theorem 3.2 b of [14]): the boundary values of √ det H σ are equal to π in the corners (0, 0) and (−1, 0) and 4π in the corner (−1/2, 1/2), and the extension inside is dictated by the harmonicity of κ 1/2 (z). The behaviors at these two types of corners are referred to as ferroelectric and anti-ferroelectric phases in [6].…”

“…Remark 1. In the example of [6], for the so-called 'scalar-product' boundary conditions, the arctic curve is found to be consisting of six arcs, each of them is a portion of certain ellipses. This can be explained via the reduction to lozenge model, as follows.…”

“…We show in the last section, section 4, that the four-vertex model of [6] also has trivial potential, and in fact its limit shapes are obtained from the lozenge dimer model limit shapes by a linear transformation.…”