Topological expansion of unitary integrals and maps

In this article, we study integrals on the unitary group with respect to the Haar measure. We give a combinatorial interpretation in terms of maps of the asymptotic topological expansion, established previously by Guionnet and Novak. The maps we introduce – the maps of unitary type – satisfy Tutte like equations. It allows us to show that in the perturbative regime they describe the different orders of the asymptotic topological expansion. Furthermore, they generalize the monotone Hurwitz numbers.