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Supplement to “Asymptotic power of sphericity tests for highdimensional data.” DOI:10.1214/13AOS1100SUPP
, 2013
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The largest eigenvalue of real symmetric, Hermitian and hermitian selfdual random matrix models with rank one external source, part I.” arXiv:1012.4144
, 2010
"... We consider the limiting location and limiting distribution of the largest eigenvalue in real symmetric (β = 1), Hermitian (β = 2), and Hermitian selfdual (β = 4) random matrix models with rank 1 external source. They are analyzed in a uniform way by a contour integral representation of the joint ..."
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We consider the limiting location and limiting distribution of the largest eigenvalue in real symmetric (β = 1), Hermitian (β = 2), and Hermitian selfdual (β = 4) random matrix models with rank 1 external source. They are analyzed in a uniform way by a contour integral representation of the joint probability density function of eigenvalues. Assuming the “oneband” condition and certain regularities of the potential function, we obtain the limiting location of the largest eigenvalue when the nonzero eigenvalue of the external source matrix is not the critical value, and further obtain the limiting distribution of the largest eigenvalue when the nonzero eigenvalue of the external source matrix is greater than the critical value. When the nonzero eigenvalue of the external source matrix is less than or equal to the critical value, the limiting distribution of the largest eigenvalue will be analyzed in a subsequent paper. In this paper we also give a definition of the external source model for all β> 0. 1 Introduction and statement of results 1.1
Limits of spiked random matrices
, 2013
"... Given a large, highdimensional sample from a spiked population, the top sample covariance eigenvalue is known to exhibit a phase transition. We show that the largest eigenvalues have asymptotic distributions near the phase transition in the rank one spiked real Wishart setting and its general β ana ..."
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Cited by 17 (2 self)
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Given a large, highdimensional sample from a spiked population, the top sample covariance eigenvalue is known to exhibit a phase transition. We show that the largest eigenvalues have asymptotic distributions near the phase transition in the rank one spiked real Wishart setting and its general β analogue, proving a conjecture of Baik, Ben Arous and Péche ́ (2005). We also treat shifted mean Gaussian orthogonal and β ensembles. Such results are entirely new in the real case; in the complex case we strengthen existing results by providing optimal scaling assumptions. One obtains the known limiting random Schrödinger operator on the halfline, but the boundary condition now depends on the perturbation. We derive several characterizations of the limit laws in which β appears as a parameter, including a simple linear boundary value problem. This PDE description recovers known explicit formulas at β = 2, 4, yielding in particular a new and simple proof of the Painleve ́ representations for these
On the largest eigenvalue of a Hermitian random matrix model with spiked external source I. Rank one case
"... Abstract Consider a Hermitian matrix model under an external potential with spiked external source. When the external source is of rank one, we compute the limiting distribution of the largest eigenvalue for general, regular, analytic potential for all values of the external source. There is a tran ..."
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Cited by 13 (4 self)
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Abstract Consider a Hermitian matrix model under an external potential with spiked external source. When the external source is of rank one, we compute the limiting distribution of the largest eigenvalue for general, regular, analytic potential for all values of the external source. There is a transitional phenomenon, which is universal for convex potentials. However, for nonconvex potentials, new types of transition may occur. The higher rank external source is analyzed in the subsequent paper.
Universality for the largest eigenvalue of sample covariance matrices with general population
, 2013
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Scaling Limits of Correlations of Characteristic Polynomials for the Gaussian βEnsemble with
 External Source, Int. Math. Res. Notices
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Hypergeometric Functions of Matrix Arguments and Linear Statistics of MultiSpiked Hermitian Matrix Models
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Painlevé II in random matrix theory and related fields. arXiv:1210.3381
, 2012
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