Topological Expansion of Oscillatory BGW and HCIZ Integrals at Strong Coupling

TL;DR

This paper proves that BGW and HCIZ integrals have large N topological expansions at strong coupling, with coefficients linked to Hurwitz numbers, revealing new connections between matrix integrals and algebraic geometry.

Contribution

It establishes the existence of topological expansions for BGW and HCIZ integrals at strong coupling with coefficients related to Hurwitz numbers, extending previous results to complex parameters.

Findings

Topological expansions exist for BGW and HCIZ integrals at strong coupling.

Expansion coefficients are genus-specific generating functions for Hurwitz numbers.

Results connect matrix integrals with algebraic geometry through Hurwitz numbers.

Abstract

We prove that the BGW and HCIZ integrals admit large N topological expansions for complex coupling and complex external fields, provided the coupling is sufficiently strong. The expansion coefficients are holomorphic functions which are genus-specific generating functions for the monotone single and double Hurwitz numbers, respectively.