Random real branched coverings of the projective line

Abstract

In this paper, we construct a natural probability measure on the space of real branched coverings from a real projective algebraic curve $(X,c_X)$ to the projective line $(\mathbb{C}{\mathbb{P}}^1,\text{conj})$. We prove that the space of degree $d$ real branched coverings having "many" real branched points (for example more than ${\sqrt{d}}^{1+\alpha }$, for any $\alpha >0$) has exponentially small measure. In particular, maximal real branched coverings, that is real branched coverings such that all the branched points are real, are exponentially rare.