| dc.description.abstract | In this thesis we study certain random walks on the two-dimensional lattice, known as
the Manhattan and Lorentz Mirror models, and certain quantum spin systems which are
generalisations of the quantum Heisenberg model. The topics are united by the fact that
we use the Brauer and walled Brauer algebras, and the representation theory of these
algebras, to study both.
We give an overview of Brauer and walled Brauer algebras, as well as that of the
symmetric group and the classical groups, and the representation theory of general finitedimensional
algebras. A key feature of the representation theory of the groups and algebras
studied in this thesis is called Schur-Weyl duality. We give an account of this theory, as
well as applying it to our work on quantum spin systems.
We study the Manhattan and Lorentz Mirror models on a cylinder of finite width. We
give an estimate on the vertical distance travelled by the walk along the cylinder, as the
cylinder width grows large. We use the Brauer algebra to depict paths of these walks
through the cylinder.
Our work on quantum spin systems is split into two parts, studying two classes of
models. The first is a class on the complete graph, and the second is an inhomogeneous
class, which includes models on the complete bipartite graph. In each case we give the
free energy, and formulae for certain magnetisation and total spin observables. We then
use these results to give formulae for points of phase transitions, as well as to describe
the phases of the models. For the complete graph models, we are able to draw phase
diagrams. | en_US |
