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Title:
Computing Conditional Eigenvector Statistics with SUSY
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Abstract:
Eigenvector statistics and their comparison with random matrix predictions have a history almost as long as the one for eigenvalues. The Porter-Thomas distribution, a particular form of the chi-square distribution, and its first moment, the inverse participation ratio (IPR), are two prominent examples. The Porter-Thomas distribution is the result of a density of all components of a uniformly distributed eigenvector on the unit sphere in the limit of large dimensions. Indeed, there are many examples of real systems and models where the Porter-Thomas prediction breaks down. However, when fixing the reference frame, say due to a Dirac picture where one has a perturbation and unperturbated part, and the energy/eigenvalue of the operator, then the chi-square distribution can be again recovered. Exactly the latter has been done by Paolo Barucca, Alexander Ossipov and me for the non-centered, double correlated Wishart ensemble that can be seen as a very general model for correlation matrices in time series analysis. Our approach has been the supersymmetry (SUSY) method something which is not restricted to our specific model. I will report on this analytical progress.
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