New scaling of Itzykson-Zuber integrals

We study asymptotics of the Itzykson-Zuber integrals in the scaling when one of the matrices has a small rank compared to the full rank. We show that the result is basically the same as in the case when one of the matrices has a fixed rank. In this way we extend the recent results of Guionnet and Maïda who showed that for the fixed rank scaling, the Itzykson-Zuber integral is given in terms of the Voiculescu's R-transform of the full rank matrix.