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

TL;DR

This paper reviews methods for evaluating integrals over the U(N) group, explores their connections to integrable systems and combinatorics, and investigates their behavior as N becomes large.

Contribution

It provides a comprehensive review and new insights into the large N limit of unitary group integrals, linking them to integrable hierarchies and combinatorial structures.

Findings

Connection between integrals and Toda lattice hierarchy

Diagrammatic expansion for large N analysis

Insights into the asymptotic behavior of log I

Abstract

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 $\log I$.