### Statistical Mechanics and Combinatorics Group

#### Group members:

Professor Anthony J. Guttmann
Professor Colin J. Thompson FAA
Dr Omar E. Foda (Associate Professor)
Dr Paul A. Pearce (Associate Professor)
Dr Nicholas C. Wormald (Associate Professor)
Dr Richard Brak (Senior Lecturer)
Dr Peter Forrester (ARC Senior Research Fellow)
Dr Aleks L. Owczarek (ARC Senior Research Fellow)
Dr Iwan Jensen (ARC Research Fellow)

##### Postdoctoral Research Fellows
Dr. Catherine Greenhill
Dr William Orrick
Dr Trevor Welsh

Henry Wong

#### Our Research Area.

Statistical mechanics involves the understanding of large complex
systems by averaging the behaviour of the individual components. For
example, one can understand the behaviour of a gas without describing
the motion of all the molecules involved, simply by knowing the type
and strength of the forces between the molecules, using the principles
of statistical mechanics. This powerful idea can be applied to many
and varied systems in the natural world and in the human arena. It was
not said lightly by a leading scientist that a well-trained
statistical mechanician can tackle any problem'' since, for example,
statistical mechanics graduates can be found working in high-end
financial organisations, in brain research or working for the Human
Genome Project. Statistical mechanics is one of the major tools in
condensed matter physics (for example it led to the understanding of
liquid crystals as seen in wrist watches) but has now been applied to
the understanding of many modern research topics. You will find
statistical mechanical theory applied to neural networks and the
theory of the brain, to the unlocking of the secrets of the human
genome and the characterisation of the behaviour of granular material.

The mathematics involved in statistical mechanics also varies widely.
Results from statistical mechanics make contributions, to name
just a few, to the theory of knots, to number theory including
generalisations of Rogers-Ramanujan identities, to optimisation
including the famous travelling salesman problem (which can occur in
airline scheduling), to analysis and importantly to combinatorics.

Many statistical mechanical problems are inherently discrete, or are
discretised for modelling convenience. The applications of statistical
mechanics then comes down to counting'' the allowed states or
configurations of the system involved and so statistical mechanics
becomes an enumerative \emph{combinatorial} problem. Combinatorics, in
general, studies enumeration and properties of discrete structures
such as graphs and walks on lattices.

Our group is engaged in the study of discrete statistical mechanical
problems and discrete structures in general. This involves
combinatorics, analysis such as asymptotics and the study of
multi-dimensional integrals, group theory and the study of algebraic
structures, as well as computer simulation and algorithm design. We
have a wide range of skills and expertise and the group represents one
of the world's \emph{leading} centres in statistical mechanics and its
applications, especially in combinatorics. If you are looking for a
challenging and diverse research experience come and try a topic from
statistical mechanics and combinatorics.

##### Prof. Tony Guttmann

I am interested in lattice based models of Statistical Mechanics and
Combinatorics, and the cross-fertilisation of the two areas. My
interests include the design of efficient algorithms for counting
problems associated with these areas, and the design of efficient and
novel numerical procedures for the analysis of the results. As these
problems include some of the oldest and most intractable problems
in mathematical physics and combinatorics, I have developed a
particular interest in devising novel mathematical methods to investigate
analytic properties of the exact solutions to these problems.

Highlights: I have edited a special issue of the journal Annals of
Combinatorics, to appear in 1999, devoted to the marriage'' of
Statistical Mechanics and Combinatorics.

This work in the above areas was recognised by the award of the 1998
Hannan Medal in Computational and Applied Mathematics by the

I have continuing collaborations with M.\ Bousquet M\'{e}lou and X.
Viennot (U. of Bordeaux) , C. Krattenthaler (U. of Vienna), A.
Sokal (N.Y.U.), A. Conway (Silicon Genetics, Calif.), S. Whittington
(U. of Toronto), L. Glasser (Clarkson U.) and D. Welsh and J. Cardy
(U. of Oxford). I work with two postdoctoral fellows Will Orrick and
Christoph Richard and currently supervise Markus Voege, Andrew
Rechnitzer and Henry Wong.

##### Prof. Colin Thompson

My research in recent years has focussed on modelling complex
systems using ideas and techniques from Statistical Mechanics and
Dynamical Systems theory. A collaborative ARC/Telstra grant
(1996-1998) for example was concerned with modelling interactions
between biological systems and electromagnetic fields. Modelling
financial markets and the behaviour of market players are the other
areas of interest where Statistical Mechanical methods are being
applied. Collaborative projects with environmental scientists on the
quantitative risk analysis, of genetically modified crops for example,
using stochastic dynamical systems are currently in progress.

I am currently working on several joint projects with Mark
Burgman (U. of Melb.), Sam Yang (CSIRO), David Bardos (DSTO), Jack
Rowley and Vitas Anderson (Telstra Res. Labs), Ben Thompson (Colben
Dynamics) and Jim McGuire (Florida State Univ.).

##### Dr Paul Pearce

The main thrust of my current research is into integrable boundary
conditions in exactly solvable lattice models in two dimensions and their
connections to boundary rational conformal field theories and perturbed
integrable field theories.

I have three major current collaborations: namely with Roger Behrend
(Stony Brook), Valentina Petkova (Sofia) and Jean-Bernard Zuber (Saclay,
Paris) on Boundary Conditions in Rational Conformal Field Theories';
with Ole Warnaar (Amsterdam) on Analytic Calculation of Cylinder
Partition Functions for ABF Models'; and with Leung Chim (Adelaide) on
TBA Excitations and Renormalization Group Flow'.

##### Dr Nick Wormald

My broad research area is combinatorics, with emphasis on asymptotic
enumeration, random graphs, graph theory and combinatorial
algorithms. The interest in asymptotic enumeration and random graphs,
which are often inter-related, is that the broad properties of numbers
of objects, or of random objects, can often be described in appealing
ways even when the precise behaviour is extremely complicated. Random
graph to be proved even when no explicit construction is known. The
techniques are also useful for analysing algorithms. Current research
includes random regular graphs, analysis of randomised algorithms
using differential equations, counting graphs which are embedded or
can be embedded in the plane, and the degrees of vertices in random
graphs. A quite separate area is the investigation of shortest
networks together with Prof.\ Rubinstein in the Geometry and Topology
group. This partly involves collaboration with companies involved in
underground mining.

I have continuing collaborations with researchers at the Courant
Institute (New York), at Microsoft, at ANU, at the University of
Georgia, at two universities in Canada (Carleton and Toronto), and at
Adam Mickiewicz University (Poland). I supervise three students
presently, Hilda Assiyatun, Billy Duckworth and Ana Jancic and
collaborate with my research fellow, Dr Ian Wanless.

##### Dr Richard Brak

My general interest lies in lattice path problems in combinatorics and
statistical mechanics. They are often problems that are simple to
state but require the most sophisticated techniques to solve. They are
used in a variety of modelling situations, including physical
chemistry applications of polymers in solution, and to computer science
applications related to computer language construction.

As an example, consider a set of paths of a certain length on a
directed square lattice starting and ending at some chosen set of
points. The paths are not allowed to have any vertices in common. How
many different configurations are there? For certain special starting
and ending positions the number of such configurations can be written
as a product form. I am trying to find combinatorial proofs of these
product forms. If the paths can have vertices in common then they are
called osculating lattice paths and are related to the six-vertex
model of magnetism and the combinatorial problem of enumerating
alternating sign matrices. I am currently characterising the
combinatorics of osculating lattice paths. I collaborate with
Professor J. Essam of the University of London on these topics as well
as local colleagues Prof.\ Tony Guttmann and Dr Aleks Owczarek. Andrew
Oppenheim (Masters student) is also working with me in this area.

##### Dr. Omar Foda

The subject of exact solutions: two-dimensional lattice models,
conformal field theories and strings, is intimately connected to
almost all aspects of classical mathematics: algebra, geometry,
complex analysis, representation theory, and many others. Recently,
connections with partition theory, Rogers-Ramanujan-type q-series
identities, Young tableaux, modular representations of the symmetric
group, and other topics in algebraic combinatorics have emerged. I am
interested in exploring all aspects of algebraic combinatorics that
appear in exact solutions. Broadly speaking, my aim is to use each
subject to learn about the other. I work together with Trevor Welsh on
these topics.

##### Dr Peter Forrester

Quantum technologies have recently become a reality with the successful
construction of devices on the mesoscopic scale. This has led to a
resurgence in the theoretical and mathematical study of quantum
mechanics on this scale. Such studies have identified universal behaviours
typical of `chaotic'' quantum motion, and mathematical models involving
random matrices, random polynomials, and most surprisingly integrable
quantum many body systems have been used to predict these behaviours.
Apart from the relevance to quantum chaos, these models have a rich
mathematical content, and this forms the basis of my research
project. I am currently collaborating with my Research Fellow Dr David
McAnally, Professor Taro Nagao of Osaka University and Dr Eric Rains
at AT&T research.

##### Dr Aleks Owczarek

The central topics of my work revolve around lattice-walk problems in
statistical mechanics and their connection to lattice models that have
exact solutions, also known as integrable models. The work may involve
mathematical analysis such as the solution of functional and
difference equations or asymptotic analysis, combinatorial analysis
involving pictorial proofs, or computer work with both symbolic and
numerical computations being made. Different models lend themselves to
different approaches and so there are opportunities to be involved in
various types of project.

Lattice-walks can be used to describe long-chain polymers, biological
membranes, and phase boundaries in fluids. On the other hand, they are
fundamental combinatorial problems that occur in many other areas such
as the theory of computer programming languages. The research involves
two main approaches: Firstly, I am developing a novel solution method
that combines techniques from traditional exactly solved lattice
models and from combinatorics. This involves the solution of
difference equations, symbolic computation and mathematical
analysis. It aims at studying the scaling (ie.\ self-similar or
fractal) behaviour of walk models. Secondly, the asymptotic behaviour
of self-attracting polymers in solution, modelled by various types of
lattice-walks, is attracting new interest. My work here is aimed at
discovering which models are suitable for describing such polymers by
systematically studying various models via computer simulation and
computer enumeration. I currently work with Professor John Essam from
the University of London, Dr Thomas Prellberg from the University of
Syracuse, New York, as well as local colleagues Dr Richard Brak and
(Prof.) Tony Guttmann and students Andrew Rechnitzer (Ph.\ D) and
Andrew Oppenheim (Masters) on these topics.

##### Dr Iwan Jensen

My general research interests are lattice models in statistical mechanics,
their connection to combinatorial problems, and
their application to the modelling of chemical and biological systems.
In particular I have been involved with the development and application
of precise numerical methods, such as exact enumerations, series expansions
and Monte Carlo simulations, for the study of lattice models. Some of the
models studied recently include classical problems of statistical mechanics,
such as lattice animals related to percolation models of fluid
flow in random media, spin systems modelling magnetic materials,
and meanders, self-avoiding walks and polygons relevant to the description
of polymers. The aim of my research is to use exact enumerations and series
expansions to further our understanding of lattice models and in particular
gain insight into the analytic properties of the as yet undiscovered exact
solutions to these problems.
My main collaborators are I. Enting from CSIRO and A. Guttmann.