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 Publish in OALib Journal ISSN: 2333-9721 APC: Only $99  Views Downloads  Relative Articles The largest sample eigenvalue distribution in the rank 1 quaternionic spiked model of Wishart ensemble Probability densities and distributions for spiked and general variance Wishart$β\$-ensembles The Beta-Wishart Ensemble The rank 1 real Wishart spiked model The rank 1 real Wishart spiked model I. Finite N analysis Hypergeometric Functions of Matrix Arguments and Linear Statistics of Multi-Spiked Hermitian Matrix Models Large Deviations of the Smallest Eigenvalue of the Wishart-Laguerre Ensemble The Smallest Eigenvalue Distribution in the Real Wishart-Laguerre Ensemble with Even Topology Universal shocks in the Wishart random matrix ensemble - 1 Maximal Invariants For Lorentz Wishart Models More...
Mathematics  2008

# Spiked Models in Wishart Ensemble

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Abstract:

The spiked model is an important special case of the Wishart ensemble, and a natural generalization of the white Wishart ensemble. Mathematically, it can be defined on three kinds of variables: the real, the complex and the quaternion. For practical application, we are interested in the limiting distribution of the largest sample eigenvalue. We first give a new proof of the result of Baik, Ben Arous and P\'{e}ch\'{e} for the complex spiked model, based on the method of multiple orthogonal polynomials by Bleher and Kuijlaars. Then in the same spirit we present a new result of the rank 1 quaternionic spiked model, proven by combinatorial identities involving quaternionic Zonal polynomials (\alpha = 1/2 Jack polynomials) and skew orthogonal polynomials. We find a phase transition phenomenon for the limiting distribution in the rank 1 quaternionic spiked model as the spiked population eigenvalue increases, and recognize the seemingly new limiting distribution on the critical point as the limiting distribution of the largest sample eigenvalue in the real white Wishart ensemble. Finally we give conjectures for higher rank quaternionic spiked model and the real spiked model.

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