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

24 pages, no figures; Version 2: Published version