A requirement of this course is that the students experiment with a random matrix problem that is of interest to them. Often these explorations take the shape of a little bit of theory and a little bit of computational experimentation. Some of you might already have some ideas that are relevant to your current research. Regardless, we thought we’d put together some ideas for projects. Feel free to adapt them
1 The eigenvalue distribution function For an N × N matrix AN , the eigenvalue distribution function 1 (e.d.f.) F AN (x) is defined as F AN (x) = Number of eigenvalues of AN ≤ x . (1) N As defined, the e.d.f. is right continuous and possibly atomic i.e. with step discontinuities at discrete points. In practical terms, the derivative of (1), referred to as the (eigenvalue) level density, is simply the
The Classical Random Matrix Ensembles The Wigner Matrix (or Hermite Ensemble) The Wigner matrices [1, 2] are often known as the Hermite or Gaussian ensembles are well studied in physics and in the book by Mehta [3]. The term Wigner matrix does not require the entries be normal, though the term Gaussian ensemble and Hermite
Random Variables and Probability Densities We assume that the reader is familiar with the most basic of facts concerning continuous random variables or is willing to settle for the following sketchy description. Samples from a (univariate or multivariate) experiment can be histogrammed either in practice or as a thought experiment. Histogramming counts
In class, we saw the connection between the so-called Hermite matrix and the semi-circular law. There is actually a deeper story that connects the classical random matrix ensembles to the classical orthogonal polynomials studied in classical texts such as [1] and more recent monographs such as
