Large-genus asymptotics of saddle connection Siegel-Veech constants

TL;DR

This paper extends the asymptotic analysis of Siegel-Veech constants, which count saddle connections on translation surfaces, to all strata and multiplicities, providing a comprehensive understanding of their behavior as genus grows.

Contribution

It generalizes existing asymptotic formulas for Siegel-Veech constants to all strata and multiplicities, enhancing the understanding of their large-genus behavior.

Findings

Asymptotics of Siegel-Veech constants are now known for all strata.

The results include constants for saddle connections between distinct zeros and loops.

Provides a unified framework for large-genus asymptotics in translation surface theory.

Abstract

Siegel-Veech constants are powerful tools for counting saddle connections on a translation surface. Their computation can be involved, most famously with recursive formulas that use intricate combinatorics or intersection theory. From these formulas, asymptotics of Siegel-Veech constants for growing genus can be extracted. We extend the known asymptotics to all strata and to all multiplicities of saddle-connections between distinct zeros and of loops.