Moments of the Hermitian Matrix Jacobi Process

Luc Deleaval () and Nizar Demni ()
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Luc Deleaval: Université Marne la Vallée
Nizar Demni: Université de Rennes 1

Journal of Theoretical Probability, 2018, vol. 31, issue 3, 1759-1778

Abstract: Abstract In this paper, we compute the expectation of traces of powers of the Hermitian matrix Jacobi process for a large enough but fixed size. To proceed, we first derive the semi-group density of its eigenvalues process as a bilinear series of symmetric Jacobi polynomials. Next, we use the expansion of power sums in the Schur polynomial basis and the integral Cauchy–Binet formula in order to determine the partitions having nonzero contributions after integration. It turns out that these are hooks of bounded weight and the sought expectation results from the integral of a product of two Schur functions with respect to a generalized beta distribution. For special values of the parameters on which the matrix Jacobi process depends, the last integral reduces to the Cauchy determinant and we close the paper with the investigation of the asymptotic behavior of the resulting formula as the matrix size tends to infinity.

Keywords: Hermitian matrix Jacobi process; Schur polynomial; Symmetric Jacobi polynomial; Hook; 15B52; 33C45; 60H15 (search for similar items in EconPapers)
Date: 2018
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DOI: 10.1007/s10959-017-0761-5

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