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Research Theme: Critical Phenomena


Theme Leader: Richard Brak (UM)
Deputy Leader: Aleks Owczarek (UM)
Other CI members: Tony Guttmann (UM), Honorary Professorial Fellow: Ian Enting (UM).

Critical phenomena are universal features of complex systems displaying emergent behaviour. This describes the phenomenon whereby macroscopic behaviour emerges from interactions between simpler, microscopic components.

Goal: Mathematical analysis of highly correlated and interacting systems.

Illustrative problem: Statistical mechanical models of critical phenomena provide key models of complex systems displaying the central features of long-range correlations and the ability to see large changes in behaviour for small parameter changes. Solving and analysing key statistical mechanical problems such as self-avoiding walks is a central motivation.

Applications: Epidemic prediction, protein and polymer design, are important applications of the work being pursued.

Work has been organised into several major projects. This was confirmed at the recent Theme Workshop, held at MASCOS, Thursday 11th October 2007. They are as follows:

• Directed walk problems and integrability in dynamical systems;
• Transfer matrix analysis;
• Exact enumeration techniques including virial expansion techniques;
• Markov Chain Monte Carlo techniques for SAW;
• Stochastic enumeration techniques;
• Distribution estimation methods;
• Quasi-Monte Carlo methods;
• Single Polymer critical phenomena;
• Magnetic systems including Ising and the Kosterlitz-Thouless transition;
• Percolating systems.

Upcoming Theme Workshops:


Introduction

It is typical of complex systems that there are situations in which they can exhibit gross changes in macroscopic properties as a result of small changes in the nature of local interactions. Such a phenomenon occurs, for example, when a statistical mechanical system undergoes a phase-change, or when an engineering system moves from stability to instability. Such critical changes are important from a practical point of view: the macroscopic properties that determine the utility of the system are often the same properties that undergo a phase change. Also, it is often necessary to answer questions about critical phenomena before an investigator can go on to characterise more complicated behaviour of a system. For example, an engineer usually needs to know whether a system is stable before its level of performance can be investigated.

Questions about critical phenomena have been approached differently by various mathematical and statistical disciplines. A researcher working in stochastic processes might investigate whether an appropriate model is recurrent or transient, a statistical mechanic may study the emergent behaviour that leads to phase changes, a control engineer will analyse stability, and a statistician might investigate the limiting behaviour of sequences of random variables. Each of these approaches is related, and often they lead to similar underlying mathematical formulations.

The mathematical techniques developed to solve problems in critical phenomena have proved to be exceptionally powerful, and hence applicable to other fields. A specific example is the method of simulated annealing, which has now become a powerful technique in Operations Research. The concept of ergodicity in statistical mechanics has proved fundamental in devising better Monte Carlo algorithms. The transfer matrix technique, first developed in statistical mechanics, has been applied to a variety of problems in condensed matter physics, theoretical chemistry and algebraic combinatorics. The notion of universality has allowed lattice models to be used to answer computationally intractable problems in lattice field theory, such as the determination of the order of the phase transition of the finite temperature four-dimensional SU(3) lattice gauge theory. An important part of the Centre's activity is the continuing development and dissemination of powerful techniques applicable to other areas of complex systems. Its co-operative nature maximizes the opportunity for such dissemination to be effective.

A further problem in critical phenomena that relates to the Centre’s other research themes occurs in the study of the change in limiting behaviour of estimators of an intensity parameter as it passes through a critical value. Well-known special cases occur in the contexts of traffic intensity in queuing processes, in epidemic thresholds, in autoregression when passing from stationary behaviour first to a random walk and then to explosive behaviour, and in the neighbourhood of critical values in branching processes. Recent work by one of the Chief Investigators (Heyde) has shown that this limit behaviour can exhibit unusual and interesting properties. A more comprehensive investigation of this phenomenon is likely to prove fruitful.

By making a study of critical phenomena one of its themes, the Centre focuses a broad variety of expertise from different disciplines onto related problems. Projects undertaken by the Centre's researchers in this area include:

Critical Phenomena — 2007-8

KOSTERLITZ-THOULESS TRANSTIONS
Ian Enting (Profesorial Fellow) and Tony Guttmann (Director)

Various two-dimensional systems have two distinct types of ordering: a discrete alignment at low temperatures, a wave-like state with long correlations at higher temperatures, and a disordered state at high temperautres. The transitions at the upper and lower limits of the wave-like state are extremely weak with essental singularities rather than the more common power law singularities. Series expansions and Monte Carlo simulations are being applied to special models where exact results provide guidance about the phase diagram, in order to identify appropriate techniques for analysing more general systems.

Critical Phenomena — 2006

TRAFFIC FLOW
Critical Phenomena Team

A model of traffic flow was developed. Vehicles enter a road at one end with some probability and exit at the other end with a different probability. This can be modelled as a Markov process. The stationary state (corresponding to a constant traffic flow) is of primary interest. Surprisingly, the stationary state can be understood as an equilibrium statistical mechanical system undergoing a phase transition. The combinatorial aspects of the stationary state, in particular its representation as a lattice path model, were studied.

Critical Phenomena — 2006

POLYMER EXTENSION AND COLLAPSE
Critical Phenomena Team

The unfolding of a polymer pulled by an external force was studied. The polymer was modelled as an interacting self-avoiding walk with a pulling force in one direction. Using exact enumeration data, force-extension curves were studied, obtaining results in good agreement with experiments. This work models a famous experiment on DNA performed at the College de France. A new class of models for polymer collapse was introduced, given by random walks on regular lattices which are weighted according to multiple site visits. This model was studied on the square and simple cubic lattices, constraining the number of visits to a site. A variant of the model forbids immediate self-reversals of the random walk. Evidence is found that the existence of a collapse transition depends sensitively on the details of the model and has an unexpected dependence on dimension.

Critical Phenomena — 2006

SELF-AVOIDING WALKS
Critical Phenomena Team

Improved algorithms for the enumeration of self-avoiding walks in three dimensions were developed, which has helped further our understanding of this famous model of polymers.

Critical Phenomena — 2006

PAVINGS AND PERCOLATION
Critical Phenomena Team

The combinatorial concept of ‘pavings’ on a path graph was extended and developed. These pavings were used to give previously unavailable expressions for orthogonal polynomials, which are also important in other areas of mathematics. The concept of pavings was also used to deepen our mathematical insight by finding a new combinatorial derivation of certain recurrences on path weight. This lattice path enumeration work also has potential application in biological and other growth contexts, wherever a tri-diagonal, tetra-diagonal or similar transfer matrix is applicable. Two-dimensional percolation was also studied. In particular, numerically accurate estimates were obtained for a special parameter, notably a universal amplitude ratio, whose measurement had been elusive for many decades.

Critical Phenomena — 2004

PERCOLATION, LATTICE WALK AND RELATED MODELS
Richard Brak (Chief Investigator), Aleks Owczarek (Chief Investigator), John Essam (U London), Peter Fox (MSc student).

Results from models related to asymmetric simple exclusion process (ASEP) stochastic models (discussed further below) have been and continue to be investigated in the context of lattice polymer and percolation models.

Critical Phenomena — 2004

SELF-AVOIDING POLYGONS
Iwan Jensen (Associate Investigator) Christoph Richard (U. Bielefeld, Germany) and Tony Guttmann (Chief Investigator)

Jensen, Richard and Guttmann have developed very efficient parallel algorithms for the enumeration of self-avoiding polygons and applied these to several two-dimensional lattices. They conjectured the exact scaling function for self-avoiding polygons. The theoretical predictions were confirmed by high quality numerical calculations based on extensive series expansions. This work is continuing in order to further investigate the properties of the scaling function and relies heavily on series expansions for the full area and perimeter generating functions.

Critical Phenomena — 2004

ANALYSIS OF BIOLOGICAL AGENTS ON THE HULLS OF WATER CRAFT
Richard Brak (Chief Investigator), Aleks Owczarek (Chief Investigator), Andrew Rechnitzer (Associate Investigator), Han Kee Gan (vacation student)

This project is being conducted in conjunction with the Defense Science Technology Organisation (DSTO) and began with a successful studentship of H. Gan over the summer 2003/2004. A program to calculate the percolation cluster structure of photographs of biofouled sample plates has been written. Further work in conjunction with the Faculty of Engineering at The University of Melbourne is currently planned.

Critical Phenomena — 2004

LATTICE ANIMALS OR POLYOMINOES
Iwan Jensen (Associate Investigator)

Lattice animals are of great inherent interest in combinatorics and statistical mechanics both in their own right and also due a close relationship with percolation. Jensen has enumerated lattice animals up to size 56 on the square lattice. He has developed a very efficient parallel version of the algorithm for this problem and has extended this investigation to the hexagonal and triangular lattices.

Critical Phenomena — 2004

HEXAGONAL SYSTEMS
Tony Guttmann (Chief Investigator), Markus Voege (Ph.D. student) and Iwan Jensen (Associate Investigator)

The number of benzenoid systems, hexagonal polyominoes and polygons has been calculated, their asymptotic growth established, and rigorous bounds on the asymptotic growth proved.

Critical Phenomena — 2004

LATTICE WALKERS
Tony Guttmann (Chief Investigator), C. Krattenthaler (Lyon), X. Viennot (Bordeaux) and Yao-Ban Chan, (Centre Ph.D Scholar)

Closed form expressions for the number of vicious and friendly walkers in a strip have been obtained, and these results have been related to traditional combinatorial structures, such as Young tableaux. Further work to solve particular cases was successfully undertaken.

Critical Phenomena — 2004

SELF-AVOIDING WALKS
Iwan Jensen (Associate Investigator), S. Caracciolo (U. Milano, Italy), Tony Guttmann (Chief Investigator), A. Pelissetto (U. Roma, Italy), Andrew Rogers (PhD student) and Alan Sokal (NYU, USA)

Jensen, Guttmann, Rogers and associates have developed algorithms for the enumeration of self-avoiding walks (a model relevant to polymer science) on various lattices. They have significantly extended the series for the square, triangular and honeycomb lattices including series for metric properties such as the end-to-end distance and radius of gyration. In collaboration they undertook a careful and extensive study of the series (supplemented by Monte Carlo simulations) of the correction-to-scaling exponent of self-avoiding walks on two-dimensional lattices.


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