Large Genus Asymptotics for Siegel-Veech Constants

TL;DR

This paper investigates the asymptotic behavior of Siegel-Veech constants in large genus limits for Abelian differentials, confirming predictions about their growth and limits in connected strata.

Contribution

It provides the first large genus asymptotics for two key Siegel-Veech constants, confirming longstanding conjectures in the field.

Findings

Saddle connection constant approaches (m_i + 1)(m_j + 1) as genus grows

Area Siegel-Veech constant tends to 1/2 in large genus limit

Results confirm predictions by Zorich and Eskin-Zorich

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 $c_{\text{sc}}^{m_i, m_j} \big( \mathcal{H} (\alpha) \big)$ 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.