Hurwitz numbers and products of random matrices

We study multimatrix models, which may be viewed as integrals of products of tau functions depending on the eigenvalues of products of random matrices. We consider tau functions of the two-component Kadomtsev–Petviashvili (KP) hierarchy (semi-infinite relativistic Toda lattice) and of the B-type KP (BKP) hierarchy introduced by Kac and van de Leur. Such integrals are sometimes tau functions themselves. We consider models that generate Hurwitz numbers $H^{\mathrm E,\mathrm F}$, where $\mathrm E$ is the Euler characteristic of the base surface and $\mathrm F$ is the number of branch points. We show that in the case where the integrands contain the product of $n>2$ matrices, the integral generates Hurwitz numbers with $\mathrm E\le2$ and $\mathrm F\le n+2$. Both the numbers $\mathrm E$ and $\mathrm F$ depend both on $n$ and on the order of the factors in the matrix product. The Euler characteristic $\mathrm E$ can be either an even or an odd number, i.e., it can 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.