Of Pfaffians, correlation functions and a geometrical triumvirate of real random matrices
by Anthony Mays
Abstract: A random matrix is, broadly speaking, a matrix with entries randomly
chosen from some distribution. In the non-random case eigenvalues can
occur anywhere in the complex plane, but, remarkably, random elements
imply predictable behaviour, albeit in a probabilistic sense.
Correlation functions are one measure of a probabilistic
characterisation and we discuss a 5-part scheme, based upon orthogonal
polynomials, to calculate the eigenvalue correlation functions. We apply
this scheme to three ensembles of random matrices, each of which can be
identified with one of the surfaces of constant Gaussian curvature: the
plane, the sphere and the anti- or pseudo-sphere. We will be using real
random matrices, which possess the added complication of having a finite
probability of real eigenvalues.
This talk aims to be accessible, and to that end we will start with a
general overview of random matrices and then discuss the 5-step method,
hopefully keeping technicalities to a minimum, and with plenty of pictures.
For More Information: contact Guoqi Qian, email: email@example.com