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Correlation Functions, Cluster Functions, and Spacing Distributions for Random Matrices

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Abstract

The usual formulas for the correlation functions in orthogonal and symplectic matrix models express them as quaternion determinants. From this representation one can deduce formulas for spacing probabilities in terms of Fredholm determinants of matrix-valued kernels. The derivations of the various formulas are somewhat involved. In this article we present a direct approach which leads immediately to scalar kernels for the unitary ensembles and matrix kernels for the orthogonal and symplectic ensembles, and the representations of the correlation functions, cluster functions, and spacing distributions in terms of them.

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Authors and Affiliations

  1. Department of Mathematics and Institute of Theoretical Dynamics, University of California, Davis, California, 95616;

    Craig A. Tracy

  2. Department of Mathematics, University of California, Santa Cruz, California, 95064;

    Harold Widom

Authors
  1. Craig A. Tracy
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  2. Harold Widom
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Tracy, C.A., Widom, H. Correlation Functions, Cluster Functions, and Spacing Distributions for Random Matrices. Journal of Statistical Physics 92, 809–835 (1998). https://doi.org/10.1023/A:1023084324803

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  • DOI: https://doi.org/10.1023/A:1023084324803

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