On some integrals over the U(N) unitary group and their large N limit
P. Zinn-Justin, J.-B. Zuber

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 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 limit of .
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