UC Berkeley

The purpose of this thesis is to present some new results about the six-vertex model and dimer model, andi in particular some aspects of the limit shapes that form in the thermodynamic limit and the partial differential equations that arise in their study. Chapter 1 provides a detailed introduction to limit shape phenomena and its mathematics. Chapter 2 reviews basic facts about the six-vertex model and dimer model and their properties in the thermodynamic limit.

Chapter 3 investigates the role of integrability of the six vertex model in the formation of limit shapes. By integrability we mean, on one hand the discrete integrability of the six-vertex model in the sense of commutative families of transfer matrices, and on the other hand the integrability in the continuum limit in the Liouville sense of Poisson commutative families of functions (integrals of motion) on the phase space. The main result is to show that the partial differential equations describing the limit shape have an infinite number of conserved quantities.

Chapter 4 explains the results of the previous chapter in a more general setting. The main result is a straightforward theorem in Hamiltonian mechanics, which gives conditions for the Poisson commutativity of two Hamiltonian functions in terms of their principal action functions. It suggests a generalization of our previous results to other integrable lattice models, for example, the generalizations of the six-vertex model related to other Lie algebras.

Chapter 5 studies the six-vertex model for a special class of weights called the stochastic six-vertex model. for which the six vertex model is closely related to interacting particle models. The main result of this chapter is the derivation for the partial differential equations for the height function of the stochastic 6-vertex model on the cylinder.

Chapter 6 studies models at the intersection of discrete geometry and statistical mechanics. The main result is the asymptotic expansion of the discrete Laplacian that approximate Laplace operators induced by a Riemannian metric. They arise as a discretization of the of the Gaussian free field and are closely related to Dimer models that can be regarded as a discretization of Dirac operators in a background metric.