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 $U\subset \Sigma$: 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 ${\epsilon }_d$ is smaller than $\exp (-C_U{\epsilon }_d^{3}d)$, 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.

To appear in Mathematische Zeitschrift