# Unitary symmetry breaking in some random matrix ensembles

*Statistical Mechanics/Combinatorics Seminar*

*by Patrick Desrosiers, Visiting Research Fellow*

*Institution:*University of Melbourne

*Date: Thu 24th August 2006*

*Time: 3:15 PM*

*Location: Sisalkraft Theatre, Architecture (Ground floor), The University of Melbourne*

*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