Large Genus Asymptotics for Siegel–Veech Constants

Abstract

In this paper we consider the large genus asymptotics for two classes of Siegel-Veech constants associated with an arbitrary connected stratum H(α) of Abelian differentials. The first is the saddle connection Siegel-Veech constant c m i ,m j sc H(α) 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 carea H(α) 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 m i ,m j sc H(α) = (m i + 1)(m j + 1) 1 + o(1) and carea H(α) = 1 2 + o(1), both as |α| = 2g − 2 tends to ∞. The former result confirms a prediction of Zorich and the latter confirms one of Eskin-Zorich in the case of connected strata.

Masur–Veech volumes and intersection theory on moduli spaces of Abelian differentials

“…When the stratum is disconnected, we also show that the theorem holds for each connected component under Assumption 6.1. We remark that as an asymptotic equality as g tends to infinity, the formula (7) for the entire stratum was previously shown in the appendix by Zorich to [3] for saddle connections of multiplicity one and by Aggarwal [4] for all multiplicities.…”

“…Another application of the volume recursion is a geometric proof of the large genus limit conjecture by Eskin and Zorich [23] for the volumes of the strata and area Siegel-Veech constants. A proof using direct combinatorial arguments was given by Aggarwal [3,4]. Our proof, in addition, gives a uniform expression for the second order term as conjectured in [42] (see Sect.…”

“…In Sect. 4 we exhibit another recursion of volumes by using the algebra of shifted symmetric functions and cumulants. In Sect.…”

Uniform Lower Bound for Intersection Numbers of ψ-Classes

“…It is widely believed that the third term of Virasoro constraints becomes negligible in large genera. We expect that technique from [1] might be useful for replacing the lower bound in (10) by the exact asymptotics under strengthening restrictions on α.…”

“…Remark 2. It is plausible, that much stronger statement might be true, where the bound n < C log(g) is replaced by the bound n < g α with any fixed α satisfying α < 1 2 . The reason why one cannot go beyond the bound n < √ g is explained at the very end of Section 1.…”