My research interests are in equilibrium statistical mechanics in general, and more particularly in discrete models of phase transitions. As these are often equivalent to a combinatorial problem, I am equally interested in the relevant combinatorics, and the connection between the two. Additionally, I am regularly looking for new ways to count the underlying graphs efficiently. This leads me into a study of algorithmic complexity.

Current Research Projects

Phase Transitions and Critical Phenomena

The susceptibility of the two-dimensional Ising model
with B. G. Nickel, W. P. Orrick and J. H. H. Perk

The susceptibility of the two-dimensional Ising model is a long-standing problem of both great interest and great difficulty in lattice statistics. We have recently given a polynomial time algorithm for the generation of series coefficients (time for N terms is proportional to N^6). As a result we have obtained hundreds of series coefficients, and given an analysis of the scaling function out to terms in t^{15}, where t = 1 - T/T_c. Confluent logarithmic terms are also identified, as is the scaling behaviour of the antiferromagnetic susceptibility. Papers to J. Stat. Phys. and Phys. Rev. Letts. are in preparation.

Polymer collapse transitions
with R. Brak, A. Owczarek

In a dilute solution, a polymer with some form of attractive inter-monomer interaction can undergo a phase transition. At high temperatures the polymer is in an extended "stretched-out" phase, but as the temperature is lowered the effects of the interaction dominate and the polymer collapses into a ball-like structure. We are studying several lattice models of polymers which undergo this phase transition. The widely studied "self-avoiding walk" model can be studied by exact enumeration and series analysis. This model represents linear polymers. Another model, a "directed" self-avoiding walk, can be solved exactly and the existence of a phase transition proved. A model of ring polymers is self-avoiding polygons. This model is also being studied by exact enumeration, series analysis and Monte Carlo simulations.

Osculating Lattice Paths and Friendly Walkers
with R.Brak, C. Krattenthaler, X. G. Viennot and M. Voege

Osculating lattice paths are sets of directed lattice paths that are not allowed to cross but may intersect. They occur in the six-vertex ice models, alternating sign matrices and provide a simple model of dense polymer systems. Friendly walkers may not only touch, but may share successive bonds of the lattice. They are related to other vertex models, and provide a natural generalisation of previously solved problems. The effect of impenetrable walls is also considered.

Lattice Enumeration Problems

Ising and Potts models
with I. Jensen and W. Orrick

We are studying the anisotropic Ising model and Potts model in two dimensions. Recent developments have shown that the anisotropic model gives considerable insight into the analytic structure of the solution for unsolved problems, such as the Ising susceptibility and Potts magnetisation and free energy. Together with Larry Glasser, of Clarkson University, we are taking a fresh look at the susceptibility of the two-dimensional Ising model.

Metric properties of self-avoiding walks
with A. Owczarek and R.Brak

For planar self-avoiding walks the correction-to-scaling exponents remain of interest. By calculating various metric properties, such as the mean-square radius of gyration, mean-square end-to-end distances, the mean distance of a monomer from the origin etc., various invariants can be constructed that are expected to shed new light on this question. As a by-product of this work, the distribution function of self-avoiding walks is also being studied. Complementary Monte Carlo work is being carried out at New York University by A.D. Sokal and his colleagues.

Directed percolation
with I. Jensen

Further work on directed percolation is being carried out. We are developing series for anisotropic directed percolation, which, in analogy with the Ising study above, gives information about the analytic structure of the solution.

Polygon problems
with A. Rechnitzer and W. Orrick

The generating function for various solvable polygon problems is being investigated in order to find an inversion relation. The aim is to use the inversion relation and any symmetries as a method of solution. Other, related polygon problems are under investigation similarly.

Inversion relations
with A. Rechnitzer and W. Orrick

The existence and behaviour of inversion relations for various combinatorial and statistical mechanical models is being studied.

Extension procedures for series expansions
with I. Jensen

The idea of finding "correction terms" to finite lattice model calculations is being exploited. This has resulted in considerably extended series for various statistical mechanics problems.

Functional approximants
with C. Richard

A new class of approximation method that reflects known asymptotic behaviour is being studied.

Polygon and walk enumeration
with I. Jensen, W. Orrick and A. Rechnitzer

Various aspects of self-avoiding walks and polygons are being investigated.

Polygons and polyominoes on the hexagonal lattice
with M. Voege

The finite-lattice method is being used to study the number of polyominoes and polygons (enumerated by area) on the hexagonal lattice. This is related to the number of benzenoid systems, much studied by chemists..

The Chiral Potts Model
with I.G. Enting, J. Perk and H. Au-Yang

We are extending series for various realisations of the chiral Potts model, in order to calculate the phase diagram. The work is proceeding slowly due to my lack of contribution.