Monotone Hurwitz numbers and the HCIZ integral I

Details Monotone Hurwitz numbers and the HCIZ integral I Monotone Hurwitz numbers and the HCIZ integral I

2011-07-06

arxiv.org/abs/1107.1015v2

Mathematics

Combinatorics

34 pages, substantial revisions, references added

Scientific paper

In this article, we study the notion of genus expansion in the Harish-Chandra-Itzykson-Zuber matrix model. We prove that, under suitable hypotheses, each Taylor coefficient of the HCIZ free energy admits an $N \rightarrow \infty$ 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. 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.

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