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An exact formula for general spectral correlation function of random Hermitian matrices

Yan V. Fyodorov·Eugene Strahov
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Abstract

We have found an exact formula expressing a general correlation function containing both products and ratios of characteristic polynomials of random Hermitian matrices. The answer is given in the form of a determinant. An essential difference from the previously studied correlation functions (of products only) is the appearance of non-polynomial functions along with the orthogonal polynomials. These non-polynomial functions are the Cauchy transforms of the orthogonal polynomials. The result is valid for any ensemble of beta=2 symmetry class and generalizes recent asymptotic formulae obtained for GUE and its chiral counterpart by different methods..

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Yan V. Fyodorov, Eugene Strahov. 2003-05-09. An exact formula for general spectral correlation function of random Hermitian matrices. https://doi.org/10.1088/0305-4470%2F36%2F12%2F320

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