Large Genus Asymptotics for Siegel-Veech Constants

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 $c_{\text{area}}(\mathcal{H}(\alpha ))$ 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)$,bothas$|\alpha| = 2g - 2$tendsto$\infty$. The former result confirms a prediction of Zorich and the latter confirms one of Eskin-Zorich in the case of connected strata.