Spectral curves for hypergeometric Hurwitz numbers

We consider multi-matrix 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 Grotendieck's dessins d'enfants) of fixed genus, degree, and the ramification profiles at two points, z1 and zn. Ramifications at other n−2 points enter the sum with the length of the profile at z2 and with the total length of profiles at the remaining n−3 points. We find the spectral curve of the model for n=5 using the loop equation technique for the above generating function represented as a chain of Hermitian matrices with a nearest-neighbor interaction of the type tr MiMi+1−1. The obtained spectral curve is algebraic and provides all necessary ingredients for the topological recursion procedure producing all-genus terms of the asymptotic expansion of our model in 1∕N2. We discuss braid-group symmetries of our model and perspectives of the proposed method.