A matrix model for hypergeometric Hurwitz numbers

We present multimatrix models that are generating functions for the numbers of branched covers of the complex projective line ramified over n fixed points zi, i = 1, ..., n (generalized Grothendieck’s dessins d’enfants) of fixed genus, degree, and ramification profiles at two points z1 and zn. We sum over all possible ramifications at the other n-2 points with a fixed length of the profile at z2 and with a fixed total length of profiles at the remaining n-3 points. All these models belong to a class of hypergeometric Hurwitz models and are therefore tau functions of the Kadomtsev-Petviashvili hierarchy. In this case, we can represent the obtained model as a chain of matrices with a (nonstandard) nearest-neighbor interaction of the type tr MiMi+1−1. We describe the technique for evaluating spectral curves of such models, which opens the way for obtaining 1/N2-expansions of these models using the topological recursion method. These spectral curves turn out to be algebraic.