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.
@article{0402073,
author = {Collins, Benoit and Sniady, Piotr},
title = {Integration with respect to the Haar measure on unitary, orthogonal and
symplectic group},
journal = {arXiv},
volume = {2004},
number = {0},
year = {2004},
language = {en},
url = {http://dml.mathdoc.fr/item/0402073}
}
Collins, Benoit; Sniady, Piotr. Integration with respect to the Haar measure on unitary, orthogonal and
symplectic group. arXiv, Tome 2004 (2004) no. 0, . http://gdmltest.u-ga.fr/item/0402073/