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Exact Wigner Surmise Type Evaluation of the Spacing Distribution in the Bulk of the Scaled Random Matrix Ensembles

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Abstract

Random matrix ensembles with orthogonal and unitary symmetry correspond to the cases of real symmetric and Hermitian random matrices, respectively. We show that the probability density function for the corresponding spacings between consecutive eigenvalues can be written exactly in the Wigner surmise type form a(s)e−b(s) for a simply related to a Painlevé transcendent and b its anti-derivative. A formula consisting of the sum of two such terms is given for the symplectic case (Hermitian matrices with real quaternion elements).

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Melbourne, Parkville, Victoria, 3052, Australia

    P. J. Forrester

  2. Department of Mathematics and Statistics, (and School of Physics), University of Melbourne, Parkville, Victoria, 3052, Australia

    N. S. Witte

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  1. P. J. Forrester
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  2. N. S. Witte
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Forrester, P.J., Witte, N.S. Exact Wigner Surmise Type Evaluation of the Spacing Distribution in the Bulk of the Scaled Random Matrix Ensembles. Letters in Mathematical Physics 53, 195–200 (2000). https://doi.org/10.1023/A:1011074616607

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