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Spacings in Orthogonal and Symplectic Random Matrix Ensembles

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Abstract

It is one of the famous features of random matrices that several local eigenvalue statistics are for growing matrix sizes asymptotically described by a universal limiting law. It is universal in the sense that it does not depend on the details of the underlying matrix ensemble but on its symmetry class only. In this paper, we consider the universality of the nearest neighbor eigenvalue spacing distribution in invariant random matrix ensembles. Focusing on orthogonal and symplectic invariant ensembles, we show that the empirical spacing distribution converges in a uniform way. More precisely, the main result states that the expected Kolmogorov distance of the empirical spacing distribution from its universal limit converges to zero as the matrix size tends to infinity.

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Acknowledgments

The author would like to thank the unknown referees and Martin Venker for helpful comments and suggestions.

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Authors and Affiliations

  1. Institute of Mathematical Statistics, University of Münster, Orléans-Ring 10, 48149, Münster, Germany

    Kristina Schubert

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  1. Kristina Schubert
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Correspondence to Kristina Schubert.

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Communicated by Arno Kuijlaars.

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Schubert, K. Spacings in Orthogonal and Symplectic Random Matrix Ensembles. Constr Approx 42, 481–518 (2015). https://doi.org/10.1007/s00365-015-9274-6

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  • DOI: https://doi.org/10.1007/s00365-015-9274-6

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