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 Riemann-Hilbert 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, 11.45-12.45pm 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.