Electronic Communications in Probability

Eigenvectors of non normal random matrices

Florent Benaych-Georges and Ofer Zeitouni

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We study the angles between the eigenvectors of a random $n\times n$ complex matrix $M$ with density $\propto \mathrm{e} ^{-n\operatorname{Tr} V(M^*M)}$ and $x\mapsto V(x^2)$ convex. We prove that for unit eigenvectors $\mathbf{v} ,\mathbf{v} '$ associated with distinct eigenvalues $\lambda ,\lambda '$ that are the closest to specified points $z,z'$ in the complex plane, the rescaled inner product \[ \sqrt{n} (\lambda '-\lambda )\langle \mathbf{v} ,\mathbf{v} '\rangle \] is uniformly sub-Gaussian, and give a more precise statement in the case of the Ginibre ensemble.

Article information

Electron. Commun. Probab., Volume 23 (2018), paper no. 70, 12 pp.

Received: 21 June 2018
Accepted: 25 September 2018
First available in Project Euclid: 12 October 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 15B52: Random matrices 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52)

random matrices eigenvectors statistics Ginibre ensemble single ring theorem

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Benaych-Georges, Florent; Zeitouni, Ofer. Eigenvectors of non normal random matrices. Electron. Commun. Probab. 23 (2018), paper no. 70, 12 pp. doi:10.1214/18-ECP171. https://projecteuclid.org/euclid.ecp/1539309734

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