On the Complex Asymptotics of the HCIZ and BGW Integrals

Abstract

In this paper, we prove a longstanding conjecture on the asymptotic behavior of a pair of oscillatory matrix integrals: the Harish-Chandra/Itzykson-Zuber (HCIZ) integral, and the Brezin-Gross-Witten (BGW) integral. The main result gives a complete asymptotic expansion of these integrals for small complex parameters. The coefficients of these asymptotic expansions are generating functions for monotone Hurwitz numbers sorted by genus.

Preliminary version, comments welcome

References in corpus (4)

• Large deviations and stochastic calculus for large random matrices
• Large Deviation Principles via Spherical Integrals
• High order expansion of matrix models and enumeration of maps
• Double Hurwitz numbers: polynomiality, topological recursion and intersection theory

Cited by in corpus (9)

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• The large N limit of SU(N) integrals in lattice models
• The tensor Harish-Chandra-Itzykson-Zuber integral I: Weingarten calculus and a generalization of monotone Hurwitz numbers
• Moment Methods on compact groups: Weingarten calculus and its applications
• Jacobi Beta Ensemble and $b$-Hurwitz Numbers
• Asymptotic Behavior of Multiplicative Spherical Integrals and S-transform
• Topological expansion of unitary integrals and maps