Integration with respect to the Haar measure on unitary, orthogonal and symplectic group

TL;DR

This paper simplifies the calculation of polynomial integrals over Haar measures on unitary, orthogonal, and symplectic groups, extending formulas to all dimensions and analyzing asymptotic behaviors for large matrices.

Contribution

It provides a simplified algebraic approach to exact polynomial integrals over these groups, extending validity to all dimensions and deriving asymptotic results.

Findings

Formulas valid for all dimensions d

Exact character expansion for orthogonal and symplectic groups

Asymptotic freeness of Haar-distributed matrices

Abstract

We revisit the work of the first named author and using simpler algebraic arguments we calculate integrals of polynomial functions with respect to the Haar measure on the unitary group U(d). The previous result provided exact formulas only for 2d bigger than the degree of the integrated polynomial and we show that these formulas remain valid for all values of d. Also, we consider the integrals of polynomial functions on the orthogonal group O(d) and the symplectic group Sp(d). We obtain an exact character expansion and the asymptotic behavior for large d. Thus we can show the asymptotic freeness of Haar-distributed orthogonal and symplectic random matrices, as well as the convergence of integrals of the Itzykson-Zuber type.