The surface of a crystal - on an atomic scale - is rough and random but at large scales it becomes smooth and faceted. The six-vertex (or square ice) model - a statistical mechanics model - displays such limit shape and facet formation. The goal of the project is to develop a detailed and exact mathematical analysis for limit shapes arising in this and related models. The challenge is to understand this complex non-homogeneous large scale behaviour from microscopic local interactions. What are the shapes of boundaries formed between the different phases? What happens at the phase transition boundary? How much the macroscopic features depend on the microscopic details of the model? The project is going to study this kind of questions through the methods of complex analysis.

2023

2024