Aspects of Spectral Statistics in the Correlated Wishart Model

When multivariate empirical time series are considered to study complex systems the correlation matrix and its eigenvalues play a central role, because they bear rich information about the dynamics of the system. In the majority of the applications of time series analysis, the length of the time series is rather short such that the correlation matrix inherits statistical fluctuations. To quantify the significance of the empirically estimated correlation matrix, it is compared to a null hypothesis. Under the assumption of Gaussian statistics, the null hypothesis turns out to be the correlated Wishart model. I study aspects of the spectral statistics in the correlated Wishart model, extend and apply the method of supersymmetry to it and develop a new approach to study the statistics of the smallest and the largest eigenvalue. I focus mainly on the extreme eigenvalues, because they carry significant system specific information. In addition I consider the statistics of the eigenvalues in the bulk. In the first two parts of this thesis I briefly motivate my approach from the physical point of view and summarize the tools and statistical quantities important later. As a first result of this thesis I extend the generalized Hubbard-Stratonovich transformation to include also correlated Wishart ensembles. In the third part I am concerned with the distribution of the smallest eigenvalue within the real and the real quaternion uncorrelated Wishart model. For the real ensemble with even rectangularity, I derive an exact expression for the distribution as well as its microscopic limit and for the real quaternion model I uncover a Pfaffian structure. In the fourth part I apply the method of supersymmetry to eigenvalue statistics of the more involved correlated Wishart and Jacobi ensemble. I study the statistics of the extreme eigenvalues in the former and derive for both quantities previously unknown invariant matrix models. I calculate an exact expression and the microscopic limit of the smallest eigenvalue distribution. In the correlated Jacobi model I compute using supersymmetry the level density and obtain for the real ensemble a twofold integral expression and for the complex ensemble a closed-form expression. After this interlude I derive an asymptotic relation between bulk eigenvalue statistics of two real Wishart models with and without a degeneracy in the empirical eigenvalues and show that the local eigenvalue fluctuations are universal. In the fifth part I develop a new approach to analyze the eigenvalue statistics in the correlated Wishart model by circumventing the Itzykson-Zuber integral. I apply it to the gap probabilities related to the smallest and the largest eigenvalue distribution and derive an exact expression for the cumulative density function of the latter. Finally, I show that in some cases the smallest and the largest eigenvalue are Tracy-Widom distributed.



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