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 Painleve equations, isomonodromy deformation of differential equations, and the Riemann-Hilbert problem. Topics of ongoing study include spacing distributions, eigenvalue distributions in the complex plane and low rank perturbations of the classical random matrix ensembles.
Macdonald polynomial theory
Over forty years ago the many body Schrodinger operator with \(1/r^2\) pair potential 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 \(1/r^2\) 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.
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.
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.
My book 'Log-gases and Random matrices' (PUP, 2010)
I started working on this project in August 1994, and finished (apart from minor changes) 15 years later. It can be browsed from its Princeton University Press web page.