Matrix integrals and Hurwitz numbers

Abstract

We study multi-matrix models which may be viewed as integrals of products of tau functions which depend on the eigenvalues of products of random matrices. In the present paper we consider tau functions of the hierarchy the two-component KP (semiinfinite relativistic Toda lattice) and of hierarchy of the BKP introduced by Kac and van de Leur. Sometimes such integrals are tau functions themselves. We consider models which generate Hurwitz numbers $H^e,f$, where $e$ is the Euler characteristic of the base surface and $f$ is the number of branch points. We show that in case the integrands contains the product of $n>2$ matrices the integral generates Hurwitz numbers with $e\le 2$ and $f\le n+2$, both numbers $e$ and $f$ depend both on $n$ and on the order of the multipliers in the matrix product. The Euler characteristic $e$ can be either an even or an odd number, that is, match both orientable and nonorientable (Klein) base surfaces, depending on the presence of the tau function of the BKP hierarchy in the integrand. We study two cases: the products of complex and the products of unitary matrices.

39 pages (and Appendices taken from previous works). A number of misprints is corrected. List of references is completed

References in corpus (6)

• Recent exact and asymptotic results for products of independent random matrices
• Differential equations for singular values of products of Ginibre random matrices
• Combinatorial solutions to integrable hierarchies
• The matrix model for hypergeometric Hurwitz numbers
• Hard edge limit of the product of two strongly coupled random matrices
• The Non-Orientable Map Asymptotics Constant $p_g$

Cited by in corpus (3)

• Hurwitz numbers and matrix integrals labeled with chord diagrams
• Integrals of tau functions
• Character expansion of matrix integrals