HCIZ integral and 2D Toda lattice hierarchy
Résumé
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.