April 9, 2004: Dimitri Gioev, CIMS
Introduction to Random Matrix Theory
We will start by explaining briefly the main physical
motivation for
Random Matrix Theory (RMT), namely that it suggests a model that
describes
the statistical behavior of energy levels of complex systems.
There are three main types of ensembles of
random matrices that are
physically motivated: unitary, orthogonal, and symplectic. After that
we define the Unitary Ensemble of random matrices, introduce the basic
probabilistic quantities of interest, and show how these quantities can
be expressed in terms of orthogonal polynomials (OP's). We will then
explain the idea of universality in RMT, and in particular introduce
the appropriate scaling limit.
Universality means that the statistical
behavior predicted by RMT
should not depend on a particular choice of distribution of the matrix
elements (which has no physical meaning), but should depend only on the
type of symmetry imposed on the ensemble (in this case, unitary) which
is
physically meaningful.
At this point it will be apparent that the
proof of universality for Unitary Ensembles reduces to a study of
asymptotics of the OP's. Such a study is possible due to the fact that
the OP's solve a certain RiemannHilbert problem (RHP). Finally, we
mention the appropriate RHP, and the proof of universality for the
unitary case.
This talk can serve as an explanation of the basic RMT and is
related to our first talk on Fri Apr 16, 11.4512.45pm at Courant. That
fist talk is on our recent joint work with P.Deift on the proof of the
Universality Conjecture
for the other two cases, that is for the Orthogonal and Symplectic
Ensembles of random matrices.
