Completing the Picture for the Smallest Eigenvalue of Real Wishart Matrices

G. Akemann, T. Guhr, M. Kieburg, R. Wegner, and T. Wirtz
Phys. Rev. Lett. 113, 250201 – Published 19 December 2014; Erratum Phys. Rev. Lett. 114, 179901 (2015)

Abstract

Rectangular real N×(N+ν) matrices W with a Gaussian distribution appear very frequently in data analysis, condensed matter physics, and quantum field theory. A central question concerns the correlations encoded in the spectral statistics of WWT. The extreme eigenvalues of WWT are of particular interest. We explicitly compute the distribution and the gap probability of the smallest nonzero eigenvalue in this ensemble, both for arbitrary fixed N and ν, and in the universal large N limit with ν fixed. We uncover an integrable Pfaffian structure valid for all even values of ν0. This extends previous results for odd ν at infinite N and recursive results for finite N and for all ν. Our mathematical results include the computation of expectation values of half-integer powers of characteristic polynomials.

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  • Received 1 September 2014

DOI:https://doi.org/10.1103/PhysRevLett.113.250201

© 2014 American Physical Society

Erratum

Erratum: Completing the Picture for the Smallest Eigenvalue of Real Wishart Matrices [Phys. Rev. Lett. 113, 250201 (2014)]

G. Akemann, T. Guhr, M. Kieburg, R. Wegner, and T. Wirtz
Phys. Rev. Lett. 114, 179901 (2015)

Authors & Affiliations

G. Akemann1, T. Guhr2, M. Kieburg1, R. Wegner1, and T. Wirtz2

  • 1Fakultät für Physik, Universität Bielefeld, D-33501 Bielefeld, Germany
  • 2Fakultät für Physik, Universität Duisburg-Essen, D-47048 Duisburg, Germany

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Issue

Vol. 113, Iss. 25 — 19 December 2014

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