# On some integrals over the U(N) unitary group and their large N limit

### Paul Zinn-Justin 1, Jean-Bernard Zuber 2

#### Journal of Physics A 36 (2003) 3173-3194

The integral over the U(N) unitary group $I=\int DU \exp\Tr A U B U^\dagger$ is reexamined. Various approaches and extensions are first reviewed. The second half of the paper deals with more recent developments: relation with integrable Toda lattice hierarchy, diagrammatic expansion and combinatorics, and on what they teach us on the large $N$ limit of $\log I$.

• 1. Laboratoire de Physique Théorique et Hautes Energies (LPTHE),
CNRS : UMR7589 – Université Paris VI - Pierre et Marie Curie – Université Paris VII - Paris Diderot
• 2. Service de Physique Théorique (SPhT),
CNRS : URA2306 – CEA : DSM/SPHT