  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.
