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Moments of the characteristic polynomial in the three ensembles of random matrices

Abstract : Moments of the characteristic polynomial of a random matrix taken from any of the three ensembles, orthogonal, unitary or symplectic, are given either as a determinant or a Pfaffian or as a sum of determinants. For gaussian ensembles comparing the two expressions of the same moment one gets two remarkable identities, one between an $n × n$ determinant and an $m × m$ determinant and another between the Pfaffian of a $2n × 2n$ anti-symmetric matrix and a sum of $m × m$ determinants.
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Madan Lal Mehta, Jean-Marie Normand. Moments of the characteristic polynomial in the three ensembles of random matrices. Journal of Physics A: Mathematical and General (1975 - 2006), IOP Publishing, 2001, 34 (22), pp.4627-4639. ⟨10.1088/0305-4470/34/22/304⟩. ⟨cea-02904765⟩

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