Counting Meromorphic Functions with Critical Points of Large Multiplicities

Abstract

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 $\CP^1$ 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 asymptotic 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.