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Universality at Weak and Strong Non-Hermiticity Beyond the Elliptic Ginibre Ensemble

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Abstract

We consider non-Gaussian extensions of the elliptic Ginibre ensemble of complex non-Hermitian random matrices by fixing the trace Tr(XX*) of the matrix X with a hard or soft constraint. These ensembles have correlated matrix entries and non-determinantal joint densities of the complex eigenvalues. We study global and local bulk statistics in these ensembles, in particular in the limit of weak non-Hermiticity introduced by Fyodorov, Khoruzhenko and Sommers. Here, the support of the limiting measure collapses to the real line. This limit was motivated by physics applications and interpolates between the celebrated sine and Ginibre kernel. Our results constitute a first proof of universality of the interpolating kernel. Furthermore, in the limit of strong non-Hermiticity, where the support of the limiting measure remains an ellipse, we obtain local Ginibre statistics in the bulk of the spectrum.

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Acknowledgements

The work was supported by the German research council DFG through Grants AK35/2-1 (G.A.), IGK “Stochastics and Real World Models” Beijing–Bielefeld (M.C.), and CRC 701 “Spectral Structures and Topological Methods in Mathematics” (M.V.), as well as by the European Research Council under the European Unions Seventh Framework Programme (FP/2007/2013)/ERC Grant Agreement No. 307074 (M.V.).

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Authors and Affiliations

  1. Faculty of Physics, Bielefeld University, P.O. Box 100131, 33501, Bielefeld, Germany

    Gernot Akemann & Milan Cikovic

  2. Department of Mathematics, Ruhr University Bochum, 44780, Bochum, Germany

    Martin Venker

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  1. Gernot Akemann
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  2. Milan Cikovic
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  3. Martin Venker
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Correspondence to Gernot Akemann.

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Communicated by P. Deift

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Akemann, G., Cikovic, M. & Venker, M. Universality at Weak and Strong Non-Hermiticity Beyond the Elliptic Ginibre Ensemble. Commun. Math. Phys. 362, 1111–1141 (2018). https://doi.org/10.1007/s00220-018-3201-1

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  • DOI: https://doi.org/10.1007/s00220-018-3201-1

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