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Abstract
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The problem of calculating the scaled limit of the joint moments of the
characteristic polynomial, and the derivative of the characteristic polynomial, for
matrices from the unitary group with Haar measure first arose in studies
relating to the Riemann zeta function in the thesis of Hughes. Subsequently,
Winn showed that these joint moments can equivalently be written as the
moments for the distribution of the trace in the Cauchy unitary ensemble
and furthermore that they relate to certain hypergeometric functions based
on Schur polynomials, which enabled explicit computations. We give a
-generalisation
of these results, where now the role of the Schur polynomials is played by the Jack polynomials.
This leads to an explicit evaluation of the scaled moments and the trace distribution for all
, subject
to the constraint that a particular parameter therein is equal to a nonnegative integer.
Consideration is also given to the calculation of the moments of the singular statistic
for the Jacobi
-ensemble.
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Keywords
circular beta ensemble, Jack polynomials, random matrix
moments
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Mathematical Subject Classification
Primary: 60B20
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Milestones
Received: 4 January 2021
Revised: 21 August 2021
Accepted: 29 September 2021
Published: 11 May 2022
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