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Edge scaling limits for a family of non-Hermitian random matrix ensembles
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  • Published: 26 February 2009

Edge scaling limits for a family of non-Hermitian random matrix ensembles

  • Martin Bender1 

Probability Theory and Related Fields volume 147, pages 241–271 (2010)Cite this article

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  • 22 Citations

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Abstract

A family of random matrix ensembles interpolating between the Ginibre ensemble of n × n matrices with iid centered complex Gaussian entries and the Gaussian unitary ensemble (GUE) is considered. The asymptotic spectral distribution in these models is uniform in an ellipse in the complex plane, which collapses to an interval of the real line as the degree of non-Hermiticity diminishes. Scaling limit theorems are proven for the eigenvalue point process at the rightmost edge of the spectrum, and it is shown that a non-trivial transition occurs between Poisson and Airy point process statistics when the ratio of the axes of the supporting ellipse is of order n −1/3. In this regime, the family of limiting probability distributions of the maximum of the real parts of the eigenvalues interpolates between the Gumbel and Tracy–Widom distributions.

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Authors and Affiliations

  1. Departement Wiskunde, Celestijnenlaan 200B, 3001, Leuven, Belgium

    Martin Bender

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  1. Martin Bender
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Correspondence to Martin Bender.

Additional information

Supported by grant KAW 2005.2008 from the Knut and Alice Wallenberg Foundation.

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Cite this article

Bender, M. Edge scaling limits for a family of non-Hermitian random matrix ensembles. Probab. Theory Relat. Fields 147, 241–271 (2010). https://doi.org/10.1007/s00440-009-0207-9

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  • Received: 04 June 2008

  • Revised: 29 January 2009

  • Published: 26 February 2009

  • Issue Date: May 2010

  • DOI: https://doi.org/10.1007/s00440-009-0207-9

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Keywords

  • Random matrices
  • Non-Hermitian
  • Extremes
  • Tracy–Widom
  • Gumbel
  • Airy
  • Poisson

Mathematics Subject Classification (2000)

  • 15A52
  • 60G70
  • 60G55
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