Large genus asymptotics for frequency of non-simple curves

TL;DR

This paper derives formulas for the frequency of non-simple curves on closed surfaces and analyzes their asymptotic behavior as the genus grows large, extending Mirzakhani's work.

Contribution

It extends Mirzakhani's frequency expressions to non-simple curves and studies their large genus asymptotics, identifying the most common curve types with fixed intersections.

Findings

Formulas for non-simple curve frequencies in large genus

Identification of most common curve types with fixed intersections

Extension of Mirzakhani's frequency expressions

Abstract

We give an expression for the frequency of non-simple curves in closed surfaces and exploit it to study relative frequencies of such curves in large genus. This extend to the case of non-simple curves Mirzakhani's expressions of frequencies in terms of Konsevitch polynomials and Delecroix-Goujard-Zograf-Zorich large genus asymptotics for those frequencies. In particular, with K fixed, we identify which types of curves with K intersections are most common.