HCIZ integral and 2D Toda lattice hierarchy

Abstract

The expression of the large $N$ Harish Chandra--Itzykson--Zuber (HCIZ) integral in terms of the moments of the two matrices is investigated using an auxiliary unitary two-matrix model, the associated biorthogonal polynomials and integrable hierarchy. We find that the large $N$ HCIZ integral is governed by the dispersionless Toda lattice hierarchy and derive its string equation. We use this to obtain various exact results on its expansion in powers of the moments.Comment: 20 pages. various minor improvements/correction

Monotone Hurwitz Numbers and the HCIZ Integral

“…In the extremal cases α = (1 d ) and α = (d), this result was previously obtained by Zinn-Justin [27] and Gewurz and Merola [12], respectively. Theorem 0.3 should be compared with the well-known explicit formula for the genus zero Hurwitz numbers…”

“…Remark 3.2. The g = 0 case of Theorem 3.1 was obtained by Zinn-Justin in [23] using the Toda lattice equations. For a combinatorial solution of the Toda equations encompassing those of Okounkov and Zinn-Justin, see [8].…”

Ramifications of Hurwitz theory, KP integrability and quantum curves

“…Here we assume that the N ×N matrices A and B are diagonal, and we normalize the Haar measure on the unitary group U(N) in such a way that dU = 1. Up to a factor that is not relevant for our computations, HCIZ integral describes a tau-function of the two-dimensional Toda lattice [39,53]. The generating function of the double monotone Hurwitz numbers is given by…”

Pfaffian and Determinantal Tau Functions

“…The choice dμ(z 1 , z 2 ) = e z 1 z 2 dz 1 dz 2 , z 1,2 ∈ R yields (see [12]) the fermionic representation for the partition function of the two-matrix model introduced in [9] (the relation of the two-matrix model and TL see in [6] and different fermionic representation see in [7]). If we take z 1,2 ∈ S 1 , we obtain the model of two unitary matrices [35,36]. [8]) the fermionic representation for the partition function of the one-matrix model (for the relation of the one-matrix model, the one-dimensional TL see in [6] and different fermionic representation see [7]).…”