arXiv subjects

Tim Wirtz

Publications and source records attributed to Tim Wirtz.

23 records · Page 2Linked to original sources

arXiv2014

Limiting Statistics of the Largest and Smallest Eigenvalues in the Correlated Wishart Model

The correlated Wishart model provides a standard tool for the analysis of correlations in a rich variety of systems. Although much is known for complex correlation matrices, the empirically much more important real case still poses substantial challenges. We put forward a new approach, which maps arbitrary statistical quantities, depending on invariants only, to invariant Hermitian matrix models. For completeness we also include the quaternion case and deal with all three cases in a unified way. As an important application, we study the statistics of the largest eigenvalue and its limiting distributions in the correlated Wishart model, because they help to estimate the behavior of large complex systems. We show that even for fully correlated Wishart ensembles, the Tracy-Widom distribution can be the limiting distribution of the largest as well as the smallest eigenvalue, provided that a certain scaling of the empirical eigenvalues holds.

math-ph↗

arXiv2014

Eigenvalue Density of the Doubly Correlated Wishart Model: Exact Results

Data sets collected at different times and different observing points can possess correlations at different times $and$ at different positions. The doubly correlated Wishart model takes both into account. We calculate the eigenvalue density of the Wishart correlation matrices using supersymmetry. In the complex case we obtain a new closed form expression which we compare to previous results in the literature. In the more relevant and much more complicated real case we derive an expression for the density in terms of a fourfold integral. Finally, we calculate the density in the limit of large correlation matrices.

math-ph↗

arXiv2014

Distribution of the Smallest Eigenvalue in Complex and Real Correlated Wishart Ensembles

For the correlated Gaussian Wishart ensemble we compute the distribution of the smallest eigenvalue and a related gap probability.We obtain exact results for the complex (β=2) and for the real case (β=1). For a particular set of empirical correlation matrices we find universality in the spectral density, for both real and complex ensembles and all kinds of rectangularity. We calculate the asymptotic and universal results for the gap probability and the distribution of the smallest eigenvalue. We use the Supersymmetry method, in particular the generalized Hubbard-Stratonovich transformation and superbosonization.

math-ph↗

arXiv2013

Distribution of the Smallest Eigenvalue in the Correlated Wishart Model

Wishart random matrix theory is of major importance for the analysis of correlated time series. The distribution of the smallest eigenvalue for Wishart correlation matrices is particularly interesting in many applications. In the complex and in the real case, we calculate it exactly for arbitrary empirical eigenvalues, i.e., for fully correlated Gaussian Wishart ensembles. To this end, we derive certain dualities of matrix models in ordinary space. We thereby completely avoid the otherwise unsurmountable problem of computing a highly non-trivial group integral. Our results are compact and much easier to handle than previous ones. Furthermore, we obtain a new universality for the distribution of the smallest eigenvalue on the proper local scale.

math-ph↗

arXiv2011

Worldsheet operator product expansions and p-point functions in AdS3/CFT2

We construct the operator product expansions (OPE) of the chiral primary operators in the worldsheet theory for strings on AdS_3 x S^3 x T^4. As an interesting application, we will use the worldsheet OPEs to derive a recursion relation for a particular class of extremal p-point correlators on the sphere. We compare our result with the corresponding recursion relation previously found in the symmetric orbifold theory on the boundary of AdS_3.

hep-th↗