The genus-zero five-vertex model

Limit shapes from harmonicity: dominos and the five vertex model

“…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].…”

“…parameterizes the tangent lines to the arctic curve as u runs over R, see [14]. The roots of s u therefore correspond to the horizontal sides of R, the roots of t u correspond to the vertical lines of R, and the roots of c u are the tangent lines through the origin.…”

“…The five vertex model [8,14] is a model of non-intersecting north-and west-going lattice paths in Z 2 . The probability of a configuration (on a finite domain) is proportional to the product of its vertex weights as given in figure 10.…”

“…One can also define and solve the model for 0 < r < 1, setting the weight of the empty vertex to be 1 − r 2 , see[8,14]. We will not consider this case here.…”

Arctic curves of the four-vertex model

“…For recent progresses in relation to nonperiodic graphs, see [20], and reference therein. Concerning models that cannot be expressed in terms of free fermions, the variational principle has been solved only for the five-vertex model [21][22][23], and for some particular cases of the stochastic six-vertex model [24,25].…”