Correlation Functions, Cluster Functions and Spacing Distributions for Random Matrices

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

1998-04-02

arxiv.org/abs/solv-int/9804004v3

J. Statistical Physics 92 (1998), 809-835.

Mathematics

Spectral Theory

22 pages. LaTeX file. Minor correction

Scientific paper

10.1023/A:1023084324803

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 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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