Moments and Cumulants of Polynomial random variables on unitary groups, the Itzykson-Zuber integral and free probability
Benoit Collins
TL;DR
This work develops an explicit algebraic framework for integrating polynomial functions over the unitary group by exploiting Weingarten calculus, symmetric-group characters, and Schur functions. It delivers exact formulas for unitary integrals, a detailed asymptotic expansion of the Weingarten function, and a cumulant-based analysis that yields new proofs and refinements of asymptotic freeness for unitary conjugations. It further analyzes the large-d behavior of the Itzykson–Zuber integral, providing a geometric (planar-graph) interpretation of the limiting cumulants and a novel scaling that connects to Voiculescu’s free cumulants and the R-transform. Together, these results illuminate the link between random unitary matrices, free probability, and matrix-model integrals, with explicit formulas and computable limits guiding applications in random matrix theory and related areas.
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.