Moments and Cumulants of Polynomial random variables on unitary groups, the Itzykson-Zuber integral and free probability

TL;DR

This paper derives explicit formulas for moments and cumulants of polynomial random variables on unitary groups, explores their asymptotic behavior, and connects these results to free probability theory and the Itzykson-Zuber integral.

Contribution

It provides explicit character and Schur function-based formulas for integrals on unitary groups and links their asymptotics to free probability and Voiculescu's R-transform.

Findings

Explicit formulas for integrals using symmetric group characters and Schur functions

Asymptotic expansions of integrals as dimension d approaches infinity

Connection of large d asymptotics to free probability theory and R-transform

Abstract

We consider integrals on unitary groups $U_d$ of the form $$\int_{U_d}U_{i_1j_1}... U_{i_qj_q}U^*_{j'_{1}i'_{1}} ... U^*_{j'_{q'}i'_{q'}}dU$$ We give an explicit formula in terms of characters of symmetric groups and Schur functions, which allows us to rederive an asymptotic expansion as $d\to\infty$. Using this we rederive and strenghthen a result of asymptotic freeness due to Voiculescu. We then study large $d$ asymptotics of matrix model integrals and of the logarithm of Itzykson-Zuber integrals and show that they converge towards a limit when considered as power series. In particular we give an explicit formula for $$\lim_{d\to\infty}\frac{\partial^n}{\partial z^n}d^{-2} \log\int_{U_d} e^{zd Tr (XUYU^*)}dU|_{z=0}$$ assuming that the normalized traces $d^{-1} Tr(X^k)$ and $d^{-1} Tr (Y^k)$ converge in the large $d$ limit. We consider as well a different scaling and relate its asymptotics to Voiculescu's R-transform.