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Spacings: An Example for Universality in Random Matrix Theory

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Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 53))

Abstract

Universality of local eigenvalue statistics is one of the most striking phenomena of Random Matrix Theory, that also accounts for a lot of the attention that the field has attracted over the past 15 years. In this paper we focus on the empirical spacing distribution and its Kolmogorov distance from the universal limit. We describe new results, some analytical, some numerical, that are contained in Schubert K (2012) On the convergence of the nearest neighbour eigenvalue spacing distribution for orthogonal and symplectic ensembles. PhD thesis, Ruhr-Universität Bochum, Germany. A large part of the paper is devoted to explain basic definitions and facts of Random Matrix Theory, culminating in a sketch of the proof of a weak version of convergence for the empirical spacing distribution σ N (see (23)).

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Acknowledgements

Both authors acknowledge support from the Deutsche Forschungsgemeinschaft in the framework of the SFB/TR 12 “Symmetries and Universality in Mesoscopic Systems”. We are grateful to Peter Forrester for useful remarks.

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Authors and Affiliations

  1. University of Bayreuth, 95440, Bayreuth, Germany

    Thomas Kriecherbauer & Kristina Schubert

Authors
  1. Thomas Kriecherbauer
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  2. Kristina Schubert
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Correspondence to Thomas Kriecherbauer .

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Editors and Affiliations

  1. Westphalian Wilhelms University Münster Institute of Mathematics and Informatics, Münster, Germany

    Gerold Alsmeyer

  2. Westphalian Wilhelms University Münster Institute for Mathematical Statistics, Münster, Germany

    Matthias Löwe

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Kriecherbauer, T., Schubert, K. (2013). Spacings: An Example for Universality in Random Matrix Theory. In: Alsmeyer, G., Löwe, M. (eds) Random Matrices and Iterated Random Functions. Springer Proceedings in Mathematics & Statistics, vol 53. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38806-4_3

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