The arctic curve of the domain-wall six-vertex model

TL;DR

This paper investigates the arctic curve in the six-vertex model with domain wall boundary conditions, proposing a conjecture for its characterization and deriving its form in the disordered regime, with applications to alternating sign matrices.

Contribution

It introduces a conjecture linking the arctic curve to root condensation in saddle-point equations and computes its form in the disordered regime, including special cases.

Findings

Arctic curve is generally non-algebraic, algebraic at root-of-unity points.

Explicit parametric form of the arctic curve in the disordered regime.

Limit shape for q-enumerated alternating sign matrices as q approaches zero.

Abstract

The problem of the form of the `arctic' curve of the six-vertex model with domain wall boundary conditions in its disordered regime is addressed. It is well-known that in the scaling limit the model exhibits phase-separation, with regions of order and disorder sharply separated by a smooth curve, called the arctic curve. To find this curve, we study a multiple integral representation for the emptiness formation probability, a correlation function devised to detect spatial transition from order to disorder. We conjecture that the arctic curve, for arbitrary choice of the vertex weights, can be characterized by the condition of condensation of almost all roots of the corresponding saddle-point equations at the same, known, value. In explicit calculations we restrict to the disordered regime for which we have been able to compute the scaling limit of certain generating function entering the saddle-point equations. The arctic curve is obtained in parametric form and appears to be a non-algebraic curve in general; it turns into an algebraic one in the so-called root-of-unity cases. The arctic curve is also discussed in application to the limit shape of $q$-enumerated (with $0<q\leq 4$) large alternating sign matrices. In particular, as $q\to 0$ the limit shape tends to a nontrivial limiting curve, given by a relatively simple equation.