Vol. 3, No. 1, 2022

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Joint moments of a characteristic polynomial and its derivative for the circular $\beta$-ensemble

Peter J. Forrester

Vol. 3 (2022), No. 1, 145–170

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 β-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 β > 0, 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 j=1N1xj for the Jacobi β-ensemble.

circular beta ensemble, Jack polynomials, random matrix moments
Mathematical Subject Classification
Primary: 60B20
Received: 4 January 2021
Revised: 21 August 2021
Accepted: 29 September 2021
Published: 11 May 2022
Peter J. Forrester
School of Mathematics and Statistics and ARC Centre of Excellence for Mathematical and Statistical Frontiers
The University of Melbourne
Parkville, VIC