## Graphical representations of Ising and Potts models: stochastic geometry of the quantum Ising model and the space-time Potts model

##### View / Open Files

##### Authors

Björnberg, Jakob Erik

##### Advisors

Grimmett, Geoffrey

##### Date

2010-02-09##### Awarding Institution

University of Cambridge

##### Author Affiliation

Department of Pure Mathematics and Mathematical Statistics

Statistical Laboratory

Gonville & Caius College

##### Qualification

Doctor of Philosophy (PhD)

##### Language

English

##### Type

Thesis

##### Metadata

Show full item record##### Citation

Björnberg, J. E. (2010). Graphical representations of Ising and Potts models: stochastic geometry of the quantum Ising model and the space-time Potts model (Doctoral thesis). https://doi.org/10.17863/CAM.16208

##### Abstract

Statistical physics seeks to explain macroscopic properties of matter in terms of microscopic interactions. Of particular interest is the phenomenon of phase transition: the sudden changes in macroscopic
properties as external conditions are varied. Two models in particular are of great interest to mathematicians, namely the Ising model of a magnet and the percolation model of a porous solid. These models in turn are part of the unifying framework of the random-cluster representation, a model for random graphs which was first studied by
Fortuin and Kasteleyn in the 1970’s. The random-cluster representation has proved extremely useful in proving important facts about the Ising model and similar models.
In this work we study the corresponding graphical framework for two related models. The first model is the transverse field quantum Ising model, an extension of the original Ising model which was introduced
by Lieb, Schultz and Mattis in the 1960’s. The second model is the space–time percolation process, which is closely related to the contact model for the spread of disease. In Chapter 2 we define the
appropriate ‘space–time’ random-cluster model and explore a range of useful probabilistic techniques for studying it. The space–time Potts model emerges as a natural generalization of the quantum Ising model. The basic properties of the phase transitions in these models are treated
in this chapter, such as the fact that there is at most one unbounded fk-cluster, and the resulting lower bound on the critical value in Z.
In Chapter 3 we develop an alternative graphical representation of the quantum Ising model, called the random-parity representation. This representation is based on the random-current representation of
the classical Ising model, and allows us to study in much greater detail the phase transition and critical behaviour. A major aim of this chapter is to prove sharpness of the phase transition in the quantum Ising
model—a central issue in the theory—and to establish bounds on some critical exponents. We address these issues by using the random-parity representation to establish certain differential inequalities, integration
of which give the results.
In Chapter 4 we explore some consequences and possible extensions of the results established in Chapters 2 and 3. For example, we determine the critical point for the quantum Ising model in Z and in ‘star-like’ geometries.

##### Keywords

Mathematics

##### Identifiers

This record's DOI: https://doi.org/10.17863/CAM.16208