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.

中文翻译：

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Siegel-Veech常数的大类渐近性

在本文中，我们考虑了两类Siegel-Veech常数的大类渐近性，它们与Abelian微分的任意连通层\（\ mathcal {H}（\ alpha）\）相关。第一个是鞍形连接Siegel–Veech常数\（c _ {\ mathrm {sc}} ^ {m_i，m_j} \ big（\ mathcal {H}（\ alpha）\ big）\）计算两个不同的鞍形连接，固定规定订单\（m_i \）和\（m_j \）的零，第二个是Siegel-Veech常数\（c _ {\ mathrm {area}} \ big（\ mathcal {H}（\ alpha）\大 ）\）计算按面积加权的最大圆柱。通过对以Masur-Veech地层体积表示这些常数的Eskin-Masur-Zorich显式公式的组合分析，以及这些体积的大类渐近性的最新结果，我们证明\（c _ {\ mathrm { sc}} ^ {m_i，m_j} \ big（\ mathcal {H}（\ alpha）\ big）=（m_i + 1）（m_j + 1）\ big（1 + o（1）\ big）\）和\（c _ {\ mathrm {area}} \ big（\ mathcal {H}（\ alpha）\ big）= \ frac {1} {2} + o（1）\），都作为\（| \ alpha | = 2g-2 \）倾向于\（\ infty \）。前者的结果证实了Zorich的预测，而后者的结果证实了在连通地层中Eskin–Zorich的预测。