Critical points of random branched coverings of the Riemann sphere

Abstract

Given a closed Riemann surface $\Sigma$ equipped with a volume form $\omega$, we construct a natural probability measure on the space ${\mathcal {M}}_d(\Sigma )$ of degree d branched coverings from $\Sigma$ to the Riemann sphere ${\mathbb {C}}{\mathbb {P}}^1.$ We prove a large deviations principle for the number of critical points in a given open set $U\subset \Sigma$, that is, given any sequence $\epsilon _d$ of positive numbers, the probability that the number of critical points of a branched covering deviates from $2d\cdot \text {Vol}(U)$ more than $\epsilon _d\cdot d$ is smaller than $\exp (-C_U\epsilon ^3_d d)$, for some positive constant $C_U$. 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.