Eigenvector and Eigenvalue Statistics: From Quantum Chaos to Time Series


Eigenvector and Eigenvalue Statistics: From Quantum Chaos to Time Series

Evan WIlliams Theatre
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T: 83446796


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
    Dr Mario Kieburg, The University of Melbourne