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 $(\mathbb{C} \mathbb {P}^1,\textit{conj} )$ . 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.

中文翻译：

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投影线的随机真实分支覆盖

在本文中，我们在实数投影代数曲线 $(X,c_X)$ 到投影线 $(\mathbb{C} \mathbb {P}^1,\文本{conj} )$ 。我们证明了具有“许多”实分支点的d度实分支覆盖空间（例如，对于任何 $\alpha>0$ ，超过 $\sqrt {d}^{1+\alpha } $ ）呈指数增长小措施。特别是，最大实分支覆盖——即所有分支点都是实数的实分支覆盖——是指数级罕见的。