Autocorrelations of characteristic polynomials for the Alternative Circular Unitary Ensemble

TL;DR

This paper derives explicit formulas for moments and ratios of characteristic polynomials in the Alternative Circular Unitary Ensemble, extending known results for the classical CUE using symmetric function theory.

Contribution

It provides new closed-form expressions for moments and ratios in the ACUE, advancing the understanding of its spectral properties compared to the CUE.

Findings

Closed formulas for high moments of characteristic polynomials in ACUE

Explicit ratios of characteristic polynomials derived for ACUE

Comparison highlights differences with classical CUE results

Abstract

We find closed formulas for arbitrarily high mixed moments of characteristic polynomials of the Alternative Circular Unitary Ensemble (ACUE), as well as closed formulas for the averages of ratios of characteristic polynomials in this ensemble. A comparison is made to analogous results for the Circular Unitary Ensemble (CUE). Both moments and ratios are studied via symmetric function theory and a general formula of Borodin-Olshanski-Strahov.