Lakeside Lectures

Speaker: Professor Peter A. Clarkson (Kent University)
Title: Painleve Equations, Orthogonal Polynomials and Random Matrices
Time: 2016-06-06 (Mon.)  14:00 - 15:00
Place: Room 202, Astro-Math. Building
Abstract: In this talk I shall discuss the relationship between the Painleve equations, orthogonal polynomials and random matrices.
The six Painleve equations were first discovered around the beginning of the twentieth century by Painleve, Gambier and their colleagues in an investigation of nonlinear second-order ordinary differential equations. During the past 40 years there has been considerable interest in the Painleve equations primarily due to the fact that they integrable equations and arise as reductions of soliton equations solvable by inverse scattering. Although first discovered from strictly mathematical considerations, the Painleve equations have arisen in a variety of important physical applications including statistical mechanics, random matrices, plasma physics, nonlinear waves, quantum gravity, quantum field theory, general relativity, nonlinear optics and fibre optics.
It is well-known that orthogonal polynomials satisfy a three-term recurrence relation. In this talk I will show that for some weights the coefficients in the recurrence relation can be expressed in terms of Hankel determinants, which are Wronskians, that also arise in the description of special function solutions of a Painleve equation. These determinants also arise as partition functions in random matrix models and more generally, I will discuss the role of Painleve equations in random matrix theory.
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