Arctic curves of the twenty-vertex model with domain wall boundaries

TL;DR

This paper computes the arctic curve of the twenty-vertex model with domain wall boundaries using the tangent method, revealing diverse shapes and phase separations, and extends the approach to related tiling models with numerical validation.

Contribution

It extends the tangent method to finite geometries of the 20V model, providing explicit parametric formulas for the arctic curve and applying the technique to related tiling models.

Findings

Explicit arctic curve expressions for 20V model with various shapes.

The arctic curve separates liquid and frozen phases.

Numerical simulations confirm the analytic predictions.

Abstract

We use the tangent method to compute the arctic curve of the Twenty-Vertex (20V) model with particular domain wall boundary conditions for a wide set of integrable weights. To this end, we extend to the finite geometry of domain wall boundary conditions the standard connection between the bulk 20V and 6V models via the Kagome lattice ice model. This allows to express refined partition functions of the 20V model in terms of their 6V counterparts, leading to explicit parametric expressions for the various portions of its arctic curve. The latter displays a large variety of shapes depending on the weights and separates a central liquid phase from up to six different frozen phases. A number of numerical simulations are also presented, which highlight the arctic curve phenomenon and corroborate perfectly the analytic predictions of the tangent method. We finally compute the arctic curve of the Quarter-turn symmetric Holey Aztec Domino Tiling (QTHADT) model, a problem closely related to the 20V model and whose asymptotics may be analyzed via a similar tangent method approach. Again results for the QTHADT model are found to be in perfect agreement with our numerical simulations.