Critical points of random branched coverings of the Riemann sphere
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Mathematische Zeitschrift
Critical points of random branched coverings of the Riemann sphere Michele Ancona1 Received: 8 May 2019 / Accepted: 26 January 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Given a closed Riemann surface equipped with a volume form ω, we construct a natural probability measure on the space Md () of degree d branched coverings from to the Riemann sphere CP1 . We prove a large deviations principle for the number of critical points in a given open set U ⊂ , that is, given any sequence d of positive numbers, the probability that the number of critical points of a branched covering deviates from 2d · Vol(U ) more than d · d is smaller than exp(−CU d3 d), for some positive constant CU . 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.
Introduction This paper is concerned with the branched coverings u : → CP1 of very large degree from a closed Riemann surface to the Riemann sphere. By the Riemann-Hurwitz formula, the number of critical points of such maps, counted with multiplicity, equals #Crit(u) = 2d + 2g − 2, where g denotes the genus of and d is the degree of the map. How do these 2d + 2g − 2 critical points distribute on , if we pick u : → CP1 at random? In order to answer the question, we first construct a probability measure on the space Md () of degree d branched coverings u : → CP1 . This probability measure is denoted by μd and it is associated with a volume form ω on of total mass 1 (that is ω = 1), which is fixed once for all. Later in the introduction we will sketch the construction of the measure μd , which we will give in details in Sect. 1.3. The distribution of the critical points of a map u ∈ Md () is encoded by the associated empirical measure which we renormalize by 2d + 2g − 2, so that its mass does not depend on d ∈ N∗ . More precisely, for any degree d branched coverings u ∈ Md (), we consider the probability measure Tu on defined by
B 1
Michele Ancona [email protected] School of Mathematical Sciences, Tel Aviv University, Tel Aviv, Israel
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M. Ancona
Tu =
1 2d + 2g − 2
δx
x∈Crit(u)
where δx stands for the Dirac measure at x. The central object of the paper is then the random variable u ∈ (Md (), μd ) → Tu ∈ Prob() which takes values in the space Prob() of probabilities on . The expected value E[Tu ] of Tu converges in the weak topology to the volume form ω of , see Theorem 2.5. It means that, for any continuous function f : → R, one has Ed [Tu ( f )] Tu ( f )dμd (u) −−−→ f ω. d→∞
u∈Md ()
The main theorem of the paper is the following large deviations estimate for the random variable Tu . Theorem 0.1 Let be a closed Riemann surface equipped with a volume form ω of mass 1. For any smooth function f ∈ C ∞ (, R) and any sequence d of positive real numbers of the form d = O(d −a ), for some a ∈ [0, 1), there exists a positive constant C such that the following inequality
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