Hurwitz numbers, Belyi pairs, Grothendieck dessins d'enfant, and matrix models

Belyi pairs are functions mapping Riemann surfaces of genus $g$ on the complex projective line with branchings at a fixed number of points (at three points for the case of original Belyi pairs and Grothendieck's dessins d'enfant corresponding to these pairs). We construct the matrix model describing this situation and more general models describing the case of $n$ branching points. All these models are tau functions of the KP hierarchy and upon some constraints on their generating functions their solutions can be attained using the topological recursion technique.