- September 10, 2010 , Friday - 15h00 (Room
P
3.10):
Linear statistics of certain random matrices and
asymptotics of perturbed Toeplitz matrices
Torsten Ehrhardt
(University of California, Santa Cruz, USA)
Abstract. The theory of Random Matrices has seen a
large development during the last decade with connections to many
different areas in Mathematics. In my talk I will elaborate, by
example, only one connection with Operator Theory. Simply put, a random
matrix is a matrix whose entries are chosen at random. It is described
by the class of matrices considered and by the underlying probability
distribution. The main interest is in the eigenvalues of the random
matrices and their asymptotics when the matrix size is large. We
consider certain ensembles of non-hermitian complex random matrices and
the linear statistics of their eigenvalues. The linear statistics is a
random variable which is a sum over some test function on the
eigenvalues. In the large n limit, the linear statistics is expected to
obey some kind of central limit theorem. The key to dealing with the
asymptotics of the linear statistics is that, under certain conditions,
their probability distribution can be expressed in terms of the
determinants of matrices are the Hadarmad of a Toeplitz and a Hankel
matrix. In the cases under consideration, the Toeplitz structure is the
dominating one, and a Limit Theorem will be established, which
resembles the classical Szego Limit Theorem for Toeplitz determinants.
However, depending on the underlying Random Matrix Ensemble, two cases
with a different kind of asymptotics are identified..
Seminars take place in Lisbon, I.S.T. -
Post Graduation Building
Webpage: http://www.math.ist.utl.pt/funcional