- September 10, 2010 , Friday - 15h00 (Room P
3.10):
Torsten Ehrhardt
(University of California, Santa Cruz, USA)
" Linear statistics
of certain random matrices and asymptotics of perturbed Toeplitz
matrices "
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..
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