Critical points of random branched coverings of the Riemann sphere

Given a closed Riemann surface equipped with a volume form., we construct a natural probability measure on the space Md( ) of degree d branched coverings from to the Riemann sphere CP1. We prove a large deviations principle for the number of critical points in a given open setU. , that is, given any sequence d of positive numbers, the probability that the number of critical points of a branched covering deviates from 2d center dot Vol(U) more than d center dot d is smaller than exp(-CU 3dd), for some positive constant CU. In particular, the probability that a covering does not have any critical point in a given open set goes to zero exponential fast with the degree.