Ext. Number:
  • Mathematical Physics
  • Enumerative combinatorics
  • Statistical Mechanics
  • Stochastic Processes
  • Exact solutions of lattice models
  • Critical Phenomena (phase transitions)
  • Special functions
  • Integrable systems
Recent Publications:
  • Finite Size Effects for Spacing Distributions in Random Matrix Theory: Circular Ensembles and Riemann Zeros (2017) more
  • Singular Values of Products of Ginibre Random Matrices (2017) more
  • Finite Size Effects for Spacing Distributions in Random Matrix Theory: Circular Ensembles and Riemann Zeros (2017) more

Selected Publications

Extra Information:
My Research Interests are:

1. Random matrices.
Random matrix theory is concerned with giving analytic statistical properties of the eigenvalues and eigenvectors of matrices defined by a statistical distribution. It is found that the statistical properties are to a large extent independent of the underlying distribution, and dependent only on global symmetry properties of the matrix. Moreover, these same statistical properties are observed in many diverse settings: the spectra of complex quantum systems such as heavy nuclei, the Riemann zeros, the spectra of single particle quantum systems with chaotic dynamics, the eigenmodes of vibrating plates, amongst other examples. Imposing symmetry constraints on random matrices leads to relationships with Lie algebras and symmetric spaces, and the internal symmetry of these structures shows itself as a relection group symmetry exhibited by the eigenvalue probability densities. The calculation of eigenvalue correlation functions requires orthogonal polynomials, skew orthogonal polynomials, deteminants and Pfaffians. The calculation of spacing distributions involves many manifestations of integrable systems theory, in particular Painlev'e equations, isomonodromy deformation of differential equations, and the Riemann-Hilbert problem. Recently ensembles of matrices interpolating between the fundamental symmery classes have been identified, and the task of computing the statistical properties of the eigenvalues, using the above techniques, is being undertaken.

2. Macdonald polynomial theory.
Over thirty years ago the many body Schrodinger operator with 1 over r squared pairwise interaction was isolated as having special properties. Around fifteen years ago families of commuting differential/difference operators based on root systems were identified and subsequently shown to underly the theory of Macdonald polynomials, which are multivariable orthogonal polynomials generalizing the Schur polynomials. In fact these commuting operators can be used to write the 1 over r squared Schrodinger operator in a factorized form, and the multivariable polynomials are essentially the eigenfunctions. This has the consequence that ground state dynamical correlations can be computed. They explicitly exhibit the fractional statistical charge carried by the elementary excitations. This latter notion is the cornerstone of Laughlin's theory of the fractional quantum Hall effect, which earned him the 1998 Nobel prize for physics. The calculation of correlations requires knowledge of special properties of the multivariable polynomials, much of which follows from the presence of a Hecke algebra structure. The study of these special structures is an ongoing project.

3. Statistical mechanics and combinatorics.
Counting configurations on a lattice is a basic concern in the formalism of equilibrium statistical mechanics. Of the many counting problems encountered in this setting, one attracting a good deal of attention at present involves directed non-intersecting paths on a two-dimensional lattice. There are bijections between such paths and Young tableaux, which in turn are in bijective correspondence with generalized permutations and integer matrices. This leads to a diverse array of model systems which relate to random paths: directed percolation, tilings, asymmetric exclusion and growth models to name a few. The probability density functions which arise typically have the same form as eigenvalue probability density functions in random matrix theory, except the analogue of the eigenvalues are discrete. One is thus led to consider discrete orthogonal polynomials and integrable systems based on difference equations. The Schur functions are fundamentally related to non-intersecting paths, and this gives rise to interplay with Macdonald polynomial theory.

4.Statistical mechanics of log-potential Coulomb systems.
The logarithmic potential is intimately related to topological charge -- for example vortices in a fluid carry a topological charge determined by the circulation, and the energy between two vortices is proportional to the logarithm of the separation. The logarithmic potential is also the potential between two-dimensional electric charges, so properties of the two-dimensional Coulomb gas can be directly related to properties of systems with topological charges. In a celebrated analysis, Kosterlitz and Thouless identified a pairing phase transition in the two-dimensional Coulomb gas. They immediately realized that this mechanism, with the vortices playing the role of the charges, was responsible for the superfluid--normal fluid transition in liquid Helium films. In my studies of the two-dimensional Coulomb gas I have exploited the fact that at a special value of the coupling the system is equivalent to the Dirac field and so is exactly solvable. This has provided an analytic laboratory on which to test approximate physical theories, and has also led to the discovery of new universal features of Coulomb systems in their conductive phase.
Past Postgraduate Supervision:
Wendy BARATTA "Problems in Macdonald polynomial theory"
Benjamin FLEMING "Determinantal processes in statistical mechanical models"
Anthony MAYS "Eigenvalue distributions in the complex plane"
Maria TSARENKO "Integrable Random Tiling Models"
Current MSc Students:
Past MSc Students:
Anthony MAYS "Combinatorial aspects of juggling"
Anne LAING "Guises of the Fibonacci sequence"
Heather LONSDALE "Robinson-Schensted-Knuth Correspondence"
Ching Yun CHANG "High precision computation of some probability distributions in random matrix theory"
David GLOVER "Equivalence of construction methods for Sturmian words"
Lauren TRUMBLE "Relating random matrix theory to queueing theory"
Katarina KOVACEVIC "Spacing distributions in some random matrix systems"
Subject(s) Currently Teaching:
MAST90031 (620-647) Enumerative Combinatorics (Semester 1, 2017)
Recent Grant History:
Year(s) Source Type Title
2014 - 2016 ARC Discovery A synthesis of random matrix theory for applications in mathematics, physics and engineering (080081)
2011 - 2013 ARC Discovery Characteristic polynomials in random matrix theory
2009 - 2011 ARC Discovery The Sakai scheme - Askey table correspondence, analogues of isomonodromy and determinantal point processes
2008 - 2010 ARC Discovery Integrable structures in models of complex systems
2009 ARC Linkage International Random matrix theory and high dimensional inference
2006 - 2008 ARC Discovery Theory and applications of hypergeometric series
2003 - 2007 ARC Discovery Mathematical studies on the statistical properties of complex systems
Academic Board member
Course advisors - Upper Undergraduate
Strategic Planning Committee
UGS Committee