Parametric Random Matrix Theory

Random Matrix Theory (RMT) is the branch of probability theory which studies the statistical properties of the eigenvalues of large matrices whose matrix elements are random numbers. The theory has numerous applications ranging from quantum physics to number theory, biology, communications theory, and mathematical finance. Parametric RMT is the extension of RMT whose subject is the study of how these eigenvalues evolve as functions of time or other parameters characterising external perturbations applied to the systems whose properties are modelled by random matrices. In effect, this is a theory about spectral curves -- the lines formed by the eigenvalues as they evolve if represented as curves in the plane whose coordinates are the value of the parameter and the magnitude of the eigenvalue. The main focus of this research project is to construct, drawing on recent advances in RMT as well as on a variety of methods from quantum field theory, a comprehensive theory describing the statistical properties of the shapes of these spectral curves, and to apply this knowledge to practical problems in wireless communications and portfolio optimisation in finance.