Asymptotic correlations with corrections for the circular Jacobi $\beta$-ensemble

TL;DR

This paper computes the leading correction terms for correlation functions at the spectrum singularity in the circular Jacobi ensemble for specific Dyson indices, revealing a derivative relationship to the leading terms.

Contribution

It provides explicit calculations of correction terms for the correlation kernel at the spectrum singularity in the circular Jacobi ensemble for $eta=1,2,4$, extending previous asymptotic analyses.

Findings

Correction terms are related to leading terms via derivatives.

Analysis involves Routh-Romanovski polynomials and hypergeometric series.

Results apply to Dyson indices $eta=1,2,4$ and even $eta$ ensembles.

Abstract

Previous works have considered the leading correction term to the scaled limit of various correlation functions and distributions for classical random matrix ensembles and their $\beta$ generalisations at the hard and soft edge. It has been found that the functional form of this correction is given by a derivative operation applied to the leading term. In the present work we compute the leading correction term of the correlation kernel at the spectrum singularity for the circular Jacobi ensemble with Dyson indices $\beta = 1,2$ and 4, and also to the spectral density in the corresponding $\beta$-ensemble with $\beta$ even. The former requires an analysis involving the Routh-Romanovski polynomials, while the latter is based on multidimensional integral formulas for generalised hypergeometric series based on Jack polynomials. In all cases this correction term is found to be related to the leading term by a derivative operation.