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We shall consider the product of complex random matrices from the independentcomplex Ginibre ensembles. The product includes complex matrices $Z_i,Z_i^\dagger, \, i = 1, \ldots, n$ and $2n$ sources (complex matrices $C_i$ and$C_i^*$). Any such product can be represented by a chord diagram that encodesthe order of the matrices in the product. We introduce the Euler characteristicE$^*$ of the chord diagram and show that the spectral correlation functions ofthe product generate Hurwitz numbers that enumerate nonequivalent branchedcoverings of Riemann surfaces of genus $g^*$. The role of sources is thegeneration of branching profiles in critical points which are assigned to thevertices of the graph drawn on the base surface obtained as a result of gluingof the $2n$-gon related to the chord diagram in a standard way. Hurwitz numbersfor Klein surfaces may also be obtained by a slight modification of the model.Namely, we consider $2n+1$ polygon and consider pairing of the extra matrix$Z_{2n+1}$ with a "Mobius" tau function. Thus, the presented matrix modelslabelled by chord diagrams generate Hurwitz numbers for any given Eulercharacteristic of the base surface and for any given set of ramificationprofiles.

Hurwitz numbers and matrix integrals labeled with chord diagrams