We study the asymptotic behavior of Masur-Veech volumes as the genus goes to infinity. We show the existence of a complete asymptotic expansion of these volumes that depends only on the genus and the number of singularities. The computation of the first term of this asymptotics expansion was a long standing problem. This problem was recently solved in by Aggarwal using purely combinatorial arguments, and then by D. Chen, M. Moeller, D. Zagier and the author using algebro-geometric insights. Our proof relies on a combination of both methods.
Finally, let us mention that A. Eskin and A. Zorich proposed a series of four conjectures about the limits of numerical invariants of strata of abelian differentials: they considered Masur-Veech volumes, area Siegel-Veech constants, and refinements of these two functions according to spin parity (see @cite_14 ). All conjectures were solved in @cite_6 , @cite_0 and @cite_9 . Using the arguments of the present text, one can show that these four functions admit asymptotic expansions in the spirit of Theorem . However, we only consider the Masur-Veech volumes to keep the presentation short and clear.
@cite_0 @cite_9 @cite_14 @cite_6
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In this paper we consider the large genus asymptotics for two classes of Siegel-Veech constants associated with an arbitrary connected stratum @math of Abelian differentials. The first is the saddle connection Siegel-Veech constant @math counting saddle connections between two distinct, fixed zeros of prescribed orders @math and @math , and the second is the area Siegel-Veech constant @math 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 @math and @math , both as @math tends to @math . The former result confirms a prediction of Zorich and the latter confirms one of Eskin-Zorich in the case of connected strata. We show that the Masur-Veech volumes and area Siegel-Veech constants can be obtained by intersection numbers on the strata of Abelian differentials with prescribed orders of zeros. As applications, we evaluate their large genus limits and compute the saddle connection Siegel-Veech constants for all strata. We also show that the same results hold for the spin and hyper-elliptic components of the strata. We state conjectures on the asymptotic behavior of the volumes of moduli spaces of Abelian differentials and their Siegel–Veech constants as genus tends to infinity. We provide certain numerical evidence, describe recent advances and the state of the art towards proving these conjectures. 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 @math of a stratum indexed by a partition @math is @math as @math tends to @math . 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 @math and @math , 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.