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Limits of spiked random matrices I
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  • Published: 30 September 2012

Limits of spiked random matrices I

  • Alex Bloemendal1 &
  • Bálint Virág2 

Probability Theory and Related Fields volume 156, pages 795–825 (2013)Cite this article

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Abstract

Given a large, high-dimensional 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 et al. (Ann Probab 33:1643–1697, 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 half-line, 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 Painlevé representations for these Tracy–Widom distributions.

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Author information

Authors and Affiliations

  1. Department of Mathematics, Harvard University, Cambridge, MA, 02138, USA

    Alex Bloemendal

  2. Departments of Mathematics and Statistics, University of Toronto, Toronto, ON, M5S 2E4, Canada

    Bálint Virág

Authors
  1. Alex Bloemendal
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  2. Bálint Virág
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Correspondence to Alex Bloemendal.

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Bloemendal, A., Virág, B. Limits of spiked random matrices I. Probab. Theory Relat. Fields 156, 795–825 (2013). https://doi.org/10.1007/s00440-012-0443-2

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  • Received: 20 September 2011

  • Revised: 27 June 2012

  • Published: 30 September 2012

  • Issue Date: August 2013

  • DOI: https://doi.org/10.1007/s00440-012-0443-2

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Mathematics Subject Classification

  • 60B20
  • Random matrices (probabilistic aspects)
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