Arctic Curves of the Six-Vertex Model on Generic Domains: The Tangent Method

Abstract

We revisit the problem of determining the Arctic curve in the six-vertex model with domain wall boundary conditions. We describe an alternative method, by which we recover the previously conjectured analytic expression in the square domain. We adapt the method to work for a large class of domains, and for other models exhibiting limit shape phenomena. We study in detail some examples, and derive, in particular, the Arctic curve of the six-vertex model in a triangoloid domain at the ice-point.

Arctic Curves of the Twenty-Vertex Model with Domain Wall Boundaries

“…This symmetry was imposed for convenience throughout the paper and guarantees that the arctic curve of the 20V model is symmetric under 180 • rotation. In the case of the 6V model with DWBC, it is easily shown that there are enough sum rules for the numbers of the different types of vertices to ensure that the symmetry of weights under edge orientation reversal can be assumed without loss of generality (see for instance [CS16]). This is no longer the case for the 20V model and non symmetric weights may lead to more general arctic curves without the 180 • rotation symmetry, a situation yet to be explored.…”

“…A number of methods were developed to locate the arctic curve for non-intersecting or osculating path problems, usually in the equivalent dimer or tiling language: these methods include the asymptotic study of bulk expectation values via the technique of the Kasteleyn operator [KO06, KO07,KOS06], or the machinery of cluster integrable systems of dimers [DFSG14,KP13]. Here we will instead recourse to so-called tangent method invented by Colomo and Sportiello [CS16] whose implementation is as follows: one of the portion of the arctic curve consists in the separation line between the liquid phase and the "empty" region, i.e. a region not visited by any path.…”

“…Other portions of the curve are then obtained by the same technique upon using other equivalent path representations of the model. Even though not fully proved at this stage except in a few cases [Agg18], the tangent method was tested successfully in various models [CS16,DFL18,DFG18,DR18,DFG19b,DFG19a] and has led to a number of new predictions as it is quite easy to implement.…”

Arctic curve of the free-fermion six-vertex model with reflecting end boundary condition

“…As we shall see below, the integral formula (27) allows one to obtain the contact point between arctic curve and the left boundary, while the function H (r) N plays a fundamental role in the derivation of an analytical expression for the curve through its generating function h N (z). In order to achieve both goals, we need the asymptotic behavior of h N (z) in the scaling limit.…”

“…We can capture the behavior (59) by analyzing the integral formula of G (r) N by means of the saddle-point method. To do this, we first require the integrand of (27) to be cast in the form e N ϑ(z) , where…”