Dr Yves J M Tourigny

Disordered systems. My main interest is in developing analytical methods for the study of localisation phenomena. This interest spills over neighbouring areas, for instance the spectral theory of operators, random matrices and random walks on Lie groups.

Ph.D. Projects. I am happy to supervise capable and motivated graduate students to work on the following projects. 

Representation Theory and products of random matrices. In recent work with Alain Comtet, Jean-Marc Luck and Christophe Texier, we were able to calculate explicitly the Lyapunov exponent of a general product of random matrices in the group SL(2,R) in a certain "scaling limit" as the matrices in the product tend to the identity element. We found that the Lyapunov exponent is expressible in terms of hypergeometric functions. In his celebrated book "Special Functions and Representation Theory", Vilenkin showed that the hypergeometric functions were intimately related to the irreducible unitary representations of SL(2,R). The aim of the project is to reformulate our findings in terms of these representations, and thus to enable the analytical treatment of other groups.

Fluctuations in impurity models. Impurity models are particular disordered systems that can be formulated in terms of products of random matrices. The behaviour of these products as the number of elements in the products grows may be described by means of a "Central Limit Theorem" where the "mean" is the Lyapunov exponent, and where the "variance" quantifies the fluctuations of the product about the "mean". In recent years, much progress has been made in developing methods for the explicit calculation of the Lyapunov exponent. The aim of the project is to generalise these tools to the calculation of other "moments".