Eigenvector statistics of the product of Ginibre matrices

Zdzisław Burda, Bartłomiej J. Spisak, and Pierpaolo Vivo
Phys. Rev. E 95, 022134 – Published 24 February 2017

Abstract

We develop a method to calculate left-right eigenvector correlations of the product of m independent N×N complex Ginibre matrices. For illustration, we present explicit analytical results for the vector overlap for a couple of examples for small m and N. We conjecture that the integrated overlap between left and right eigenvectors is given by the formula O=1+(m/2)(N1) and support this conjecture by analytical and numerical calculations. We derive an analytical expression for the limiting correlation density as N for the product of Ginibre matrices as well as for the product of elliptic matrices. In the latter case, we find that the correlation function is independent of the eccentricities of the elliptic laws.

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  • Received 1 November 2016

DOI:https://doi.org/10.1103/PhysRevE.95.022134

©2017 American Physical Society

Physics Subject Headings (PhySH)

Statistical Physics & Thermodynamics

Authors & Affiliations

Zdzisław Burda* and Bartłomiej J. Spisak

  • AGH University of Science and Technology, Faculty of Physics and Applied Computer Science, al. Mickiewicza 30, 30-059 Kraków, Poland

Pierpaolo Vivo

  • Department of Mathematics, King's College London, Strand WC2R 2LS, London, United Kingdom

  • *zdzislaw.burda@agh.edu.pl
  • bjs@agh.edu.pl
  • pierpaolo.vivo@kcl.ac.uk

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Issue

Vol. 95, Iss. 2 — February 2017

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