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

Benoit Collins

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 o…