Eigenvector and Eigenvalue Statistics: From Quantum Chaos to Time Series
Evan WIlliams Theatre
Peter Hall Building
In Quantum Chaos it is a well-known fact that the components of an
eigenvector of a Hermitian Hamiltonian corresponding to a completely
chaotic system is distributed along the Porter-Thomas distribution.
Fortunately, this distribution also results from the Haar measure of one
of the corresponding classical groups which manifests the
Bohigas-Giannoni-Schmidt conjecture not only for the spectrum of
completely chaotic systems but also on the random matrix level. Despite
of that, reality is much more complicated. A general system is seldom
completely chaotic nor integrable. It is a mixture which cannot only be
seen in the statistics of the eigenvalues but the eigenvectors, as well.
This also holds for time series analysis where random matrices serve as
a bench mark model. The true underlying system specific correlations
play here the role of the integrable system and the white noise replaces
the chaotic part. Time series analysis and quantum chaotic systems are
only two of a rich variety of applications of random matrix theory.
These two examples share even one particular problem. Those eigenvalues,
which correspond to system specific information, lie inside the bulk of
the spectrum and are thus overshadowed by the ``white'' noise, they can
only be distinguished from the other eigenvalues via their eigenvectors.
In a joint work with Paolo Barucca and Alexander Ossipov, we derived the
asymptotic statistics of the components of the eigenvectors conditioned
on the considered eigenvalue and found the corrections to the
Porter-Thomas distribution, that were not known before. I am going to
give a brief introduction into this topic and highlight our main result and its significance in solving the problem described above.
Dr Mario Kieburg, The University of Melbourne