Asymptotics of numbers of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials

By A. Eskin, A. Okounkov. 2000

226 citations recorded by the source corpus.

We compute the asymptotics of the number of connected branched coverings of a torus as their degree goes to infinity and the ramification type stays fixed. These numbers are equal to the volumes of the moduli spaces of pairs (curve, holomorphic differential) with fixed multiplicities of zeros of the differential and have several applications in ergodic theory.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/2a6b2ba2f21f624d042988a89eded2139d6a4cda

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In Applied Mathematics

1. A. Zorich
2. Martin Moeller
3. C. Ulcigrai
4. Maryam Mirzakhani
5. Martin Möller
6. Dawei Chen
7. Carlos Matheus
8. G. Forni

In Geometry and Topology

1. Martin Möller
2. Martin Moeller
3. Carlos Matheus
4. G. Forni
5. A. Zorich
6. Matt Bainbridge
7. Dawei Chen
8. Maryam Mirzakhani

In Mathematical Physics

1. Martin Möller
2. Martin Moeller
3. Dawei Chen
4. A. Zorich
5. Carlos Matheus
6. G. Forni
7. Matt Bainbridge
8. Maryam Mirzakhani

Citations in the corpus

• Masur–Veech volumes and intersection theory on moduli spaces of Abelian differentials (2019, methodology): “Note that we follow the volume normalization in [EMZ03] that differs slightly from the one in [EO01] (see [CMZ18, Section 19] for the conversion).”
• Gromov-Witten theory, Hurwitz theory, and completed cycles (2002, unknown): “In particular, [11] contains the asymptotic analysis of this function as q → 1, which corresponds to the d → ∞ asymptotics of the GW-invariants.”
• SQUARE-TILED CYCLIC COVERS (2010, background): “In other words, we get a square-tiled surface, see [10, 41] (also called an origami, [25, 32]; also called an arithmetic translation surface, see [19]).”
• Fuchsian groups, coverings of Riemann surfaces, subgroup growth, random quotients and random walks (2004, unknown): “See also [13], where asymptotic results are proved in the case where the Ci consist of cycles of bounded length, and used to study volumes of certain moduli spaces.”
• Beyond Sturmian sequences: coding linear trajectories in the regular octagon (2011, methodology): “†These permutations are analogous to the ones used to describe glueing patterns in [9] or [20].”
• Euler characteristics of Teichmüller curves in genus two (2006, background): “Volumes of orbit closures and their applications to dynamics are studied in Eskin and Okounkov [24], Eskin, Okounkov and Pandharipandhe [25], Eskin, Masur and Schmoll [22], Eskin, Masur and Zorich [23], Hubert and Lelièvre [35], Lelièvre and Royer [47], and Lelièvre [46].”
• Teichmüller curves in genus two: Discriminant and spin (2005, background): “See [EMS] and [HL] for additional results on the curves Wd2, and [V1], [V2], [Vo], [Wa], [KS], [Pu], [GJ], [ EO ], and [Ca] for more on Teichm¨ uller curves.”
• Quasimodularity and large genus limits of Siegel-Veech constants (2016, methodology): “The precise version of this idea is the following formula by Eskin and Okounkov ([18, Formula 3.2]) vol (ΩMg(1n)) = lim D→∞ D−(2n+1) D∑ d=1 N0d (Tr n) .”
• Stein’s method and Plancherel measure of the symmetric group (2003, background): “Character ratios on transpositions also appear in work on the moduli space of curves [ EO ].”