MASCOS Workshop on Stochastics and Special Functions

The University of Queensland
Friday 22nd May 2009

General   Special functions crop up in all branches of mathematics, and in almost all areas of application. They include the Gamma and Beta functions, Bessel, Elliptic and Hypergeometric functions, and (famously) the Riemann zeta function. There are classes of special functions whose members are orthogonal (the inner product of distinct members is zero). Orthogonal functions include spherical harmonics and Walsh functions, but arguably the most important are the systems of orthogonal polynomials, which include Chebyshev, Hermite, Jacobi, Laguerre and Legendre polynomials.

This workshop, sponsored by the ARC Centre of Excellence for Mathematics and Statistics of Complex Systems (MASCOS), concentrated on the theory and applications of special functions with particular emphasis on how they arise in stochastic processes. There were two main speakers: Erik van Doorn, who spoke on birth-death processes and extreme zeros of orthogonal polynomials, and Peter Forrester, who spoke on the connection between the zeros of the Riemann zeta function and eigenvalues of random matrices. Additionally, there were five shorter invited presentations.

Main speakers

• Erik van Doorn (University of Twente)
• Peter Forrester (University of Melbourne)

Invited speakers

• Richard Brak (University of Melbourne)
• Jan de Gier (University of Melbourne)
• Paul Leopardi (Australian National University)
• Peter Taylor (University of Melbourne)
• Ole Warnaar (University of Queensland)

Venue   The Riverview Room, Emmanuel College, St Lucia Campus, University of Queensland [College map]

Organizers   Phil Pollett and Ole Warnaar (MASCOS and the University of Queensland)

Programme  

	09:45	Registration
	10:00	Ole Warnaar		(q,t)-Laguerre polynomials
	10:30	Break		[Refreshments provided]
	11:00	Erik van Doorn		On birth-death processes and extreme zeros of orthogonal polynomials
	12:00	Peter Taylor		The role of orthogonal polynomials in determining decay rates of multidimensional queueing processes
	12:30	Lunch Break		[Lunch provided]
	13:30	Peter Forrester		Zeros of the Riemann zeta function, eigenvalues of random matrices and queueing
	14:30	Paul Leopardi		Polynomial interpolation on the unit sphere, reproducing kernels and random matrices
	15:00	Break		[Refreshments provided]
	15:30	Jan de Gier		The asymmetric exclusion process and Askey-Wilson polynomials
	16:00	Richard Brak		Combinatorial method for calculating perturbed Tchebycheff polynomials
	16:30	Close

Abstracts

• Richard Brak

  Combinatorial method for calculating perturbed Tchebycheff polynomials

  Abstract: I will give a lightning overview of the combinatorial formulation of classical orthogonal polynomials. This understanding will then be applied to the problem of computing perturbed Tchebycheff polynomials. These polynomials satisfy the same three-term recurrence as Tchebycheff polynomials except for a finite set (the perturbation) of the weights. These polynomials have applications in polymer phase transitions and in computing the stationary state of the Asymmetric Simple Exclusion Markov Process.

• Erik van Doorn

  On birth-death processes and extreme zeros of orthogonal polynomials

  Abstract: The decay rate of a Markov process is an important quantity that characterises the speed of convergence of the time-dependent state probabilities to their limiting values. In the specific setting of a birth-death process the decay rate can be identified with the smallest point, or smallest point but one, in the support of the spectral measure of the process. The latter is the orthogonalising measure for a sequence of polynomials satisfying a three-terms recurrence relation with coefficients that are determined by the parameters of the birth-death process.

  During the last decade several new properties (bounds and positivity criteria) of the decay rate of a birth-death process have been obtained by exploiting techniques that do not involve orthogonal polynomials. In the talk I will show how these results can be translated to yield new information on the smallest and largest zeros of orthogonal polynomials, and on the support of the orthogonalising measure for an orthogonal polynomial sequence, in terms of the coefficients in the three-terms recurrence relation.

• Peter Forrester

  Zeros of the Riemann zeta function, eigenvalues of random matrices and queueing

  Abstract: The statistical properties of prime numbers have attracted the attention of many famous mathematicians. Riemann introduced what is now referred to as the Riemann zeta function for this purpose. That all the complex zeros of the Riemann zeta function lie on a certain line is the celebrated Riemann hypothesis. The computation of these Riemann zeros has attracted a lot of attention, and a combination of analytic and numerical calculations has revealed that the large zeros have statistical properties which coincide with those a large Hermitian random matrix. The latter are known in the theory of chaotic quantum systems, giving weight to a spectral interpretation of the Riemann zeros. Another surprising setting relating to the eigenvalues of random Hermitian matrices is the distribution of exit times for a fixed number of jobs begin processed by a large number of servers. The purpose of this talk is to introduce these topics, and to highlight what aspects of random matrix theory are relevant to their study.

• Jan De Gier

  The asymmetric exclusion process and Askey-Wilson polynomials

  Abstract: I will show how the steady state of the stochastic one-dimensional exclusion process with boundary reservoirs can be computed analytically in terms of Askey-Wilson polynomials. Special cases can be treated using simpler degenerate polynomials such as the Al-Salam-Chihara and q-Hermite polynomials. I will further show how time-dependent quantities can be calculated using q-Pochhammer symbols.

• Paul Leopardi

  Polynomial interpolation on the unit sphere, reproducing kernels and random matrices

  Abstract: This talk describes work in progress.

  The setting of Sloan and Womersley for polynomial interpolation on the unit sphere gives rise to a sequence of random Gram matrices. A random Gram matrix for interpolation with degree of exactnesss t is determined by (t+1)² independently uniformly distributed points on the sphere. Each entry of the matrix is given by the evaluation of a kernel polynomial at the inner product of a pair of these points. The kernel polynomials are scaled Jacobi polynomials, which vary with the degree of exactness, but converge to a function related to a Bessel function.

  Relevant questions pertain to the distribution of eigenvalues and the distribution of the determinant for a given finite degree, as well as asymptotics of the eigenvalue distribution as the degree approaches infinity.

• Peter Taylor

  The role of orthogonal polynomials in determining decay rates of multidimensional queueing processes

  Abstract: In determining the set of possible decay rates of the stationary distribution of a multidimensional queueing system we need to determine the values of x for which a system of difference equations of the form

  (1)             a(x) w^n-1 + b(x) wⁿ + c(x) wⁿ⁺¹,

  with n ≥ 0, have positive solutions in l₁. Here a(x), b(x) and c(x) are polynomials in x. Conditions for the solution of equation (1) to be in l₁ follow from elementary considerations. In joint work with Dirk Kroese, Werner Scheinhardt and Allan Motyer, I have used orthogonal polynomials to derive conditions for positivity.

  In this talk, I shall describe the context in which equation (1) arises, and then go on to discuss how we ensure positivity of the solutions.

• Ole Warnaar

  (q,t)-Laguerre polynomials

  Abstract: The (generalised) Laguerre polynomials are an important class of polynomials, orthogonal on the positive half-line with respect to the weight e^x x^a. They arise as solutions of Laguerre's differential equation and play an important role in Gaussian quadrature. In this talk I will describe a multivariable quantum extension of the Laguerre polynomials by considering solutions of an n-dimensional system of q-difference equations.