Large Genus Asymptotics for Siegel-Veech Constants
| Autor: | Aggarwal, Amol |
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| Rok vydání: | 2018 |
| Předmět: | |
| Zdroj: | Geom. Funct. Anal. 29, 1295-1324, 2019 |
| Druh dokumentu: | Working Paper |
| Popis: | In this paper we consider the large genus asymptotics for two classes of Siegel-Veech constants associated with an arbitrary connected stratum $\mathcal{H} (\alpha)$ of Abelian differentials. The first is the saddle connection Siegel-Veech constant $c_{\text{sc}}^{m_i, m_j} \big( \mathcal{H} (\alpha) \big)$ counting saddle connections between two distinct, fixed zeros of prescribed orders $m_i$ and $m_j$, and the second is the area Siegel-Veech constant $c_{\text{area}} \big( \mathcal{H}(\alpha) \big)$ counting maximal cylinders weighted by area. By combining a combinatorial analysis of explicit formulas of Eskin-Masur-Zorich that express these constants in terms of Masur-Veech strata volumes, with a recent result for the large genus asymptotics of these volumes, we show that $c_{\text{sc}}^{m_i, m_j} \big( \mathcal{H} (\alpha) \big) = (m_i + 1) (m_j + 1) \big( 1 + o(1) \big)$ and $c_{\text{area}} \big( \mathcal{H}(\alpha) \big) = \frac{1}{2} + o(1)$, both as $|\alpha| = 2g - 2$ tends to $\infty$. The former result confirms a prediction of Zorich and the latter confirms one of Eskin-Zorich in the case of connected strata. Comment: 24 pages, no figures; Version 2: Published version |
| Databáze: | arXiv |
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