Dr Yves Tourigny
B.Sc.(Montreal), M.Sc.(Waterloo), Ph.D.(Dund.)
Current positions
Senior Lecturer in Numerical Analysis
School of Mathematics
Contact
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Research interests
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".
Projects and supervisions
Research projects
VIRTUAL INTERFACE CENTRE - LINK RB1292
Principal Investigator
Managing organisational unit
School of MathematicsDates
01/01/2007 to 01/01/2010
COMPUTATIONAL NUMBER THEORY AND NUMERICAL ANALYSIS
Principal Investigator
Managing organisational unit
School of MathematicsDates
01/09/2005 to 01/09/2006
SERIES SUMMATION AND RANDOM CONTINUED FRACTIONS
Principal Investigator
Managing organisational unit
School of MathematicsDates
01/08/2004 to 01/01/2008
MOVING MESH MULTILEVEL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
Principal Investigator
Managing organisational unit
School of MathematicsDates
01/10/1997 to 01/10/2000
Publications
Recent publications
24/06/2022Generalized Lyapunov exponent for the one-dimensional Schrödinger equation with Cauchy disorder
Physical Review E
Impurity models and products of random matrices
Stochastic Processes and Random Matrices
Explicit formulae in probability and in statistical physics
In Memoriam Marc Yor--Seminaire de Probabilites XLVII
One-Dimensional Disordered Quantum Mechanics and Sinai Diffusion with Random Absorbers
Journal of Statistical Physics
Lyapunov exponents, one-dimensional Anderson localisation and products of random matrices
Journal of Physics A: Mathematical and Theoretical