Mathematics

The course aims at introducing the notion of Random MAtrices and the analysis of their spectral statistical properties. We will study the classical Wigner ensemble with the proof of the celebrated Wigner semicircle law for the eigenvalues. We wil then move on to the definition of more general Unitary Ensembles (where the underlying symmetry is given by the Unitery group) and prove fundamental structural results of Dyson on how to relate their statistical properties to the study of orthogonal polynomials. We will conclude with the study of general properties and the many applications of the theory of orthogonal polynomials, with hints at the asymptotic analysis and how this provides the first proof of Universality in the bulk and at the edge of the spectrum.