Abstract
We consider random matrices, belonging to the groups U(n), O(n) , SO(n), and Sp(n) and distributed according to the corresponding unit Haar measure. We prove that the moments of traces of powers of the matrices coincide with the moments of certain Gaussian random variables if the order of moments is low enough. Corresponding formulas, proved partly before by various methods, are obtained here in the framework of a unique method, reminiscent of the method of correlation equations of statistical mechanics. The equations are derived by using a version of the integration by parts.
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Communicated by J.L. Lebowitz
Acknowledgement We are grateful to Prof. Z. Rudnick for drawing our attention to the problem and for stimulating discussions.
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Pastur, L., Vasilchuk, V. On the Moments of Traces of Matrices of Classical Groups. Commun. Math. Phys. 252, 149–166 (2004). https://doi.org/10.1007/s00220-004-1231-3
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DOI: https://doi.org/10.1007/s00220-004-1231-3