Large Genus Asymptotics for Siegel–Veech Constants

Abstract

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 $$c_{\mathrm{sc}}^{m_i, m_j} \big ( \mathcal {H} (\alpha ) \big )$$ 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 $$c_{\mathrm{area}} \big ( \mathcal {H}(\alpha ) \big )$$ 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 $$c_{\mathrm{sc}}^{m_i, m_j} \big ( \mathcal {H} (\alpha ) \big ) = (m_i + 1) (m_j + 1) \big ( 1 + o(1) \big )$$ and $$c_{\mathrm{area}} \big ( \mathcal {H}(\alpha ) \big ) = \frac{1}{2} + o(1)$$ , 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.