Monotone Hurwitz numbers and the HCIZ Integral

We prove that the free energy of the Harish-Chandra-Itzykson-Zuber matrix model admits an N → ∞ asymptotic expansion in powers of N −2 whose coefficients are generating functions for a desymmetrized version of the double Hurwitz numbers, which we call monotone double Hurwitz numbers. Thus, the HCIZ free energy expands as a generating function enumerating certain branched covers of the Riemann sphere with arbitrary branching over 0 and ∞ and simple branching elsewhere. We prove that the monotone double Hurwitz numbers exhibit the main structural properties of the usual double Hurwitz numbers: their total generating function is a solution of the 2D Toda Lattice equations, and the numbers themselves are piecewise polynomial functions on pairs of partitions.