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Moments of the eigenvalue densities and of the secular coefficients of \(\beta\)-ensembles. (English) Zbl 1362.15027

Summary: We compute explicit formulae for the moments of the densities of the eigenvalues of the classical \(\beta\)-ensembles for finite matrix dimension as well as the expectation values of the coefficients of the characteristic polynomials. In particular, the moments are linear combinations of averages of Jack polynomials, whose coefficients are related to specific examples of Jack characters.

MSC:

15B52 Random matrices (algebraic aspects)
05E05 Symmetric functions and generalizations
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)

Software:

MOPS
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References:

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