Large Genus Asymptotics for Volumes of Strata of Abelian Differentials

Abstract

In this paper we consider the large genus asymptotics for Masur-Veech volumes of arbitrary strata of Abelian differentials. Through a combinatorial analysis of an algorithm proposed in 2002 by Eskin-Okounkov to exactly evaluate these quantities, we show that the volume ${\nu }_1({\mathcal{H}}_1(m))$ of a stratum indexed by a partition $m=(m_1,m_2,\dots ,m_n)$ is $(4+o(1)){\prod }_{i=1}^n(m_i+1{)}^{-1}$ as $2g-2={\sum }_{i=1}^n{m}_i$ tends to $\infty$. This confirms a prediction of Eskin-Zorich and generalizes some of the recent results of Chen-Moeller-Zagier and Sauvaget, who established these limiting statements in the special cases $m=1^{2g-2}$ and $m=(2g-2)$, respectively. We also include an Appendix by Anton Zorich that uses our main result to deduce the large genus asymptotics for Siegel-Veech constants that count certain types of saddle connections.

44 pages, 5 figures. Version 2: Added appendix by Anton Zorich; Version 3: Added references, expanded appendix, and more detailed exposition