Limit shapes in two-dimensional lattice models arising from physics and combinatorics

• Main Author: Keating, David
• Other Authors: Reshetikhin, Nicolai
• Format: Thesis
• Language: English
• Published: eScholarship, University of California 2021
• Subjects: Mathematics; Arctic
• Online Access: https://escholarship.org/uc/item/67j1b2kx; https://escholarship.org/content/qt67j1b2kx/qt67j1b2kx.pdf

In this thesis we study aspects of the limit shape phenomenon for two-dimensional lattice models. The three models that will be of greatest interest to us are the six vertex model, the dimer model on the hexagonal lattice, and bounded lecture hall tableaux. Chapter 1 presents a brief overview of the these objects and their relations to one another, as well as the techniques we will use to study them. In Chapter 2, we give a more detailed description of the six vertex model, and describe the Bethe ansatz method for computing the free energy in the thermodynamic limit. We show that there is an infinite family of commuting Hamiltonians governing the evolution of the limiting height function of the inhomogeneous six vertex model on a cylinder. In Chapter 3, we describe the Kasteleyn theory for dimer models on the hexagonal lattice. This dimer model can be seen as a degeneration of the six vertex model from the previous chapter. We study the asymptotic behavior of the dimer correlation functions. In Chapter 4, we turn to the study of an object arising from combinatorics known as bounded lecture hall tableaux. We show that these can be seen as a lattice model of non- intersecting paths on a certain graph. Equivalently, we show how they can be described as a dimer model. We study limit shape formation in the non-intersecting lattice path setting and conjecture formula for the arising Arctic curves. Throughout this thesis many numerical simulations of lattice models are presented. In Chapter 5, we describe an algorithm for numerically simulating lattice models utilizing the parallel processing capabilities of graphical processing units. This method of simulation applies to the previous models studied in this thesis, as well as to many other two-dimensional lattice models. Several examples are given.