Geodesic
Counting meromorphic functions with critical points of large multiplicities
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part VII, Tome 292 (2002), pp. 92-119

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We study the number of meromorphic functions on a Riemann surface with given critical values and prescribed multiplicities of critical points and values. When the Riemann surface is $\mathbb CP^1$ and the function is a polynomial, we give an elementary way of finding this number. In the general case, we show that, as the multiplicities of critical points tend to infinity, the asymptotics for the number of meromorphic functions is given by the volume of some space of graphs glued from circles. We express this volume as a matrix integral. 
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     author = {D. Panov and D. Zvonkine},
     title = {Counting meromorphic functions with critical points of large multiplicities},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {92--119},
     publisher = {mathdoc},
     volume = {292},
     year = {2002},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2002_292_a5/}
}
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D. Panov; D. Zvonkine. Counting meromorphic functions with critical points of large multiplicities. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part VII, Tome 292 (2002), pp. 92-119. http://geodesic.mathdoc.fr/item/ZNSL_2002_292_a5/