# Critical points of random branched coverings of the Riemann sphere

## TL;DR

This paper constructs a natural probability measure on branched coverings of a Riemann surface and proves a large deviations principle for the number of critical points, showing they concentrate around a predictable value as degree increases.

## Contribution

It introduces a natural probability measure on the space of branched coverings and establishes a large deviations principle for critical points, revealing their typical distribution behavior.

## Key findings

- Probability of no critical points in a set decreases exponentially with degree.
- Number of critical points concentrates around 2d times the volume of the set.
- Large deviations bound quantifies fluctuations of critical points.

## 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 $2d\cdot\textrm{Vol}(U)$ more than $\epsilon_d$ 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.