# Peter Forrester

# Home

Email: pjforr@unimelb.edu.au

Phone: +61 (0)3 8344 9683

School of Mathematics and Statistics, The University of Melbourne, Parkville, Vic 3010, Australia.

## Research Interests

### 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 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.

## Log-Gases and Random Matrices (PUP, 2010, 808pp)

I started working on this project in August 1994, and apart from minor corrections finished 15 years later.

#### A list of some corrections:

# PhD students and theses

## Wendy Baratta (2008-2010)

### Special function aspects of Macdonald Polynomial theory

#### Abstract:

The nonsymmetric Macdonald polynomials generalise the symmetric Macdonald polynomials and the nonsymmetric Jack polynomials. Consequently results obtained in nonsymmetric Macdonald polynomial theory specialise to analogous results in the theory of these other polynomials. The converse, however, does not always apply. Thus some properties of symmetric Macdonald, and nonsymmetric Jack, polynomials have no known nonsymmetric Macdonald polynomial analogues in the existing literature. Such examples are contained in the theory of Jack polynomials with prescribed symmetry and the study of Pieri-type formulas for the symmetric Macdonald polynomials. The study of Jack polynomials with prescribed symmetry considers a special class of Jack polynomials that are symmetric with respect to some variables and antisymmetric with respect to others. The Pieri-type formulas provide the branching coefficients for the expansion of the product of elementary symmetric function and a symmetric Macdonald polynomial. In this thesis we generalise both studies to the theory of nonsymmetric Macdonald polynomials.

In relation to the Macdonald polynomials with prescribed symmetry, we use our new results to prove a special case of a constant term conjecture posed in the late 1990s. A highlight of our study of Pieri-type formulas is the development of an alternative viewpoint to the existing literature on nonsymmetric interpolation Macdonald polynomials. This allows for greater extensions, and more simplified derivation, than otherwise would be possible. As a consequence of obtaining the Pieri-type formulas we are able to deduce explicit formulas for the related generalised binomial coefficients.

A feature of the various polynomials studied within this thesis is that they allow for explicit computation. Having this knowledge provides an opportunity to experimentally seek new properties, and to check our working in some of the analytic work. In the last chapter a computer software program that was developed for these purposes is detailed.

#### Published papers:

Pieri type formulas for nonsymmetric Macdonald polynomials, International Mathematics Research Notices **2009**, 2829-2854 (2009)

Some properties of Macdonald polynomials with prescribed symmetry, Kyushu Journal of Mathematics **64**, 323-343 (2010)

Further Pieri type formulas for nonsymmetric Macdonald polynomials, Journal of Algebraic Combinatorics **36**, 45-66 (2012)

## Ben Fleming (2008-2011)

### Particle system realisations of determinantal processes

#### Abstract:

Special classes of non-intersecting or interlacing particle systems, inspired by various statistical models, and their description in terms of determinantal correlation functions are the main themes of this thesis. Another unifying aspect of our subject matter is that the particle systems permit scaling limits in which their join law becomes identical to the joint law of the eigenvalues for certain ensembles of random matrices.

Particle systems resulting from a queueing model, as well as particle systems relating to tilings of hexagons, are reviewed from the viewpoint of the joint PDF and their scaling to random matric forms. An interlacing particle system relating to a limit of an \( a \times b \times c \) hexagon with \( a \) large is introduced, and forms the majority of the subject matter for this thesis. The particle system is analyzed by the computation of single line PDFs and correlation functions, as well as density profiles and scaled correlation functions in certain scaled limits.

This in turn is possible due to their being an underlying determinantal structure to the joint PDFs which carries to the correlations themselves. The functional forms obtained involve classical orthogonal polynomials, and their asymptotic properties allow scaled limits to be calculated. The scaled functional forms exhibit a universality property, being common (mostly) to a class of models which exhibit fluctuations known from random matrix theory.

Finally, a particle system relating to tilings of the Aztec diamond is studied from the viewpoint of the joint PDF, allowing seemingly original methods to be used to show known results such as the total number of possible tilings of an Aztec diamond, and the so-called arctic circle boundary.

#### Published papers:

A finitization of the bead process (with P.J. Forrester and E. Nordenstam), Probability Theory and Related Fields DOI: 10.1007/s00440-010-0324-5

Interlaced particle systems and tilings of the Aztec diamond (with P.J. Forrester), Journal of Statistical Physics **141**, 441-459 (2011)

## Anthony Mays (2007-2011)

### A geometrical triumvirate of real random matrices

#### Abstract:

The eigenvalue correlation functions for random matrix ensembles are fundamental descriptors of the statistical properties of these ensembles. In this work we present a five-step method for the calculation of these correlation functions, based on the method of (skew-) orthogonal polynomials. This scheme systemises existing methods and also involves some new techniques. By way of illustration we apply the scheme to the well known case of the Gaussian orthogonal ensemble, before moving on to the real Ginibre ensemble. A genealising parameter is then introduced to interpolate between the GOE and the real Ginibre ensemble. These real matrices have orthogonal symmetry, which is known to lead to Pfaffian or quaternion determinant processes, yet Pfaffian and quaternion determinants are not defined for odd-sized matrices. We present two methods for the calculation of the correlation functions in this case: the first is an extension of the even method, and the second establishes the odd case as a limit of the even case.

Having demonstrated our methods by reclaiming known results, we move on to study an ensemble of matrices \( Y = A^{-1} B \), where \( A \) and \( B \) are each real Ginibre matrices. This ensemble is known as the real spherical ensemble . By a convenient fractional linear transformation, we map the eigenvalues into the unit disk to obtain a rotationally invariant distribution of eigenvalues. The correlation functions are then calculated in terms of these new variables by means of finding the relevant skew orthogonal polynomials. The expected number real eigenvalues is computed, as is the probability of obtaining any number of real eigenvalues; the latter is compared to numerical simulation. The generating function is given by an explicit factorised polynomial, in which the zeros are gamma functions.

We show that in the limit of large matrix dimension, the eigenvalues (after stereographic projection) are uniformly distributed on the sphere, a result which is part of a universality result called the spherical law . By taking a different limit, we also show that the local behaviour of the eigenvalues matches that of the real Ginibre ensemble, which corresponds to the planar limit of the sphere.

Lastly, we examine the third ensemble in the triumvirate, the real truncated ensemble , which is formed by truncating \( L \) rows and columns from an \( N \times N \) Haar distributed orthogonal matrix. By applying the five-step scheme and by averaging over characteristic polynomials we proceed to calculate correlation functions and probabilities analogously to the other ensembles considered in this work. The probabilities of obtaining real eigenvalues are again compared to numerical simulation. In the large \(N\) limit (with small \(L\)) we find that the eigenvalues are uniformly distributed over the anti-sphere (after being suitably projected). This leads to a conjecture that, analogous to the circular law and spherical law, there exists an anti-spherical law. As we found for the spherical ensemble, we also find that in some limits the behaiour of this ensemble matches that of the real Ginibre ensemble.

#### Published papers:

A method to calculate correlation functions for \(\beta = 1\) random matrices of odd size (with P.J. Forrester), Journal of Statistical Physics **134**, 443-462 (2011)

Pfaffian point process for the Gaussian real generalized eigenvalue problem (with P.J. Forrester), Probability Theory and Related Fields **154**, 1-47 (2012)

## Allan Trinh (2018-2021)

### Finite N corrections in random matrix theory

#### Abstract:

Many limit laws arise from the spectral theory of large random matrices. Complementary to this is the establishment of their rate of convergence. For example, the global scaled spectral density for a general class of large and cnetred \( N \times N \) real Wigner matrices is given by the Wigner semi-circular law with a rate of convergence \( O(1/N) \). A contribution of this thesis is to perform explicit calculations for the next leading finite \( N \) corrections, which are adapted from known results on this class of random matrices.

Limit laws also apply to the soft and hard edge scaling regimes at the spectrum edge. For classical unitary ensembles, such as the Gaussian and Laguerre ensembles, spectral statistics and their corresponding finite \( N \) corrections in soft and hard edge scaling variables follow from the asymptotic results derived from orthogonal polynomial theory. Earlier studies have shown the optimal rate of convergence is \( O(1/N^{2/3}) \) at the soft edge and \( O(1/N^2) \) at the hard edge. This problem is taken up to the \( \beta \) generalisation of the Gaussian and Laguerre ensembles. Using multidimensionsal integral formulas and the generalised hypergeometric functions based on Jack polynomial theory, calculations from the spectral density for even \( \beta \) show that, with a modified soft/hard edge scaling, the optimal rate of convergence -- in keeping with the unitary case -- is \( O(1/N^2) \) at the hard edge. In the case of the Laguerre \( \beta \) ensemble for general \( \beta > 0 \), similar methods are applied to evaluate finite \( N \) corrections to the distribution of the smallest eigenvalue for \( a \in \mathbb Z_{\ge 0} \), which again show an optimal rate of convergence \( O(1/N^2) \).

In our final study, we compute the leading correction term of the correlation kernel at the spectrum singularity for the circular Jacobi ensemble. For the classical Dyson indices \( \beta = 1,2,4 \), with use of the Routh-Romanovski polynomials, we show that this correction term is related to the leading term by a derivative operation. Using methods of analysis based on multi-dimensional hypergeometric functions, this structural property of the large \( N \) asymptotic expansion is shown to hold for the spectral density with \( \beta \) even. This implies that the optimal rate of convergence is \( O(1/N^2 )\), as found at the hard edge.

#### Published papers:

Comment on "Finite size effects in the averaged eigenvalue density of Wigner random-sign real symmetric matrices" (with P.J. Forrester), Phys. Rev. E **99**, 036101 (2019)

Optimal soft edge scaling variables for the Gaussian and Laguerre \( \beta \) ensembles (with P.J. Forrester), Nucl. Phys. B **938**, 621-639 (2019)

Finite size corrections at the hard edge for the Laguerre \( \beta \) ensemble (with P.J. Forrester), Stud. Appl. Math. **143**, 315-336 (2019)

Asymptotic correlations with corrections for the circular Jacobi \( \beta \) ensemble (with P.J. Forrester and S.-H. Li), J. Approx. Theory. **271**, 105633 (2021)

# Honours Students and projects 2005-2010

## Ching Yun(Yang Ling) Chang (2005)

High precision computation of some probability distributions in random matrix theory

## Heather Dornom(Lonsdale) (2005)

Robinson-Schensted-Knuth correspondence

## Anne Laing (2005)

Guises of the Fibonacci sequence

## David Glover (2006)

Equivalence of construction methods for Sturmian words

## Anthony Mays (2006)

Combinatorial aspects of juggling

## Lauren Trumble (2006)

Relating random matrix theory to queueing theory

## Katarina Kovacevic (2008)

Spacing distributions in random matrix systems