Random Matrix Theory (MAST90103)

Random matrix theory is a diverse mathematical tool. It draws together ideas from linear algebra, multivariate calculus, analysis, probability theory, group and representation theory, differential geometry, combinatorics and mathematical physics. It also enjoys a wide number of applications, ranging from wireless communication in engineering, to time series analysis in statistics, quantum chaos and quantum field theory in physics, to the Riemann zeta function zeros and prime numbers in number theory. A self contained development of random matrix theory will be undertaken in this course from various viewpoints.

Topics to be covered include:

• Gaussian random matrix models and their application in likelihood analysis and modelling covariance matrices in time series analysis;
• eigenvalue densities and the concept of eigenvalue repulsion;
• classification of random matrices ensembles;
• derivation of Jacobians for matrix transformations such as diagonalisations;
• joint eigenvalue densities and correlation functions;
• orthogonal polynomials and the concept of determinantal point processes;
• supersymmetry and non-linear sigma-models;
• the log-gas picture;
• free probability theory and its application to matrix sums and products.