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

Abstract

We consider integrals on unitary groups $U_d$ of the form $${\int }_{U_d}{U}_{i_1{j}_1}...U_{i_q{j}_q}{U}_{j_1^{\prime }{i}_1^{\prime }}^{\ast }...U_{j_{q^{\prime }}^{\prime }{i}_{q^{\prime }}^{\prime }}^{\ast }dU$$ 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 $$\underset{d\to \infty }{\lim }\frac{{\partial }^n}{\partial z^n}{d}^{-2}\log {\int }_{U_d}{e}^{zdTr(XUY{U}^{\ast })}dU{|}_{z=0}$$ 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.

34 pages, 1 figure

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