On arctic curve phenomena in the six-vertex and related models

Details

Supervisors

Walmsley Hagendorf, Christian

Faculty

Faculté des sciences

Degree label

Master [120] en sciences physiques, à finalité approfondie

Abstract

The six-vertex model is a statistical system used to describe two-dimensional ice. The vertices of the model must satisfy the so called ice rule. With the choice of domain-wall boundary conditions, the ice rule restricts the possible configurations. A random configuration possesses ordered regions filled with one type of vertex. This phenomenon becomes increasingly apparent as the number of vertices grows. When the density of vertex tends to infinity, the border of the ordered regions becomes a curve called the arctic curve. After a first part on the definitions and properties of the six-vertex model, the thesis presents the computation of this arctic curve using the tangent method. The six-vertex model is related to various models of statistical physics. One of such models is the set of three-colourings on a grid. Numerical simulations suggest that this model also exhibits an arctic curve phenomenon with domain-wall boundary conditions. We follow Hietala's PhD thesis to explore the relationship between the partition function of the three-colour model with a reflecting end and that of the eight-vertex solid-on-solid model. The partition function of the latter model is similar to the one of the six-vertex model, where it is possible to compute the arctic curve via the tangent method. Furthermore, we study another specialization of the eight-vertex solid-on-solid model.