Unitary symmetry breaking in some random matrix ensembles
by Patrick Desrosiers, Visiting Research Fellow
Abstract: The usual ensembles of random Hermitian matrices are invariant under unitary transformations. Orthogonal polynomial theory is a powerful tool for analysing these models, especially in the scaling limit. When a
source term (external field) is added to such ensembles, the unitary symmetry is broken. This requires the introduction of polynomials that satisfy orthogonality conditions with respect to more than one weight
function. These mathematical objects are called multiple (biorthogonal)polynomials. In this seminar, I will describe how multiple polynomials
can be used in the study of correlation functions. The focus will be on the perturbation of unitary ensembles by a source term of finite rank.
The effect of this â€œsoft symmetry breakingâ€ on correlation functions will
be described by calculating asymptotics of multiple polynomials which involve generalizations of Airy and Bessel functions.
For More Information: Dr. Iwan Jensen Phone: +61 3 8344 5214 I.Jensen@ms.unimelb.edu.au