Length partition of random multicurves on large genus hyperbolic surfaces

TL;DR

This paper investigates the length distribution of multicurve components on large genus hyperbolic surfaces, showing that as genus increases, the distribution converges to a Poisson-Dirichlet law with specific length proportions for the longest components.

Contribution

It proves the convergence of length statistics of multicurves to a Poisson-Dirichlet distribution as genus tends to infinity, extending previous fixed-genus results.

Findings

Length statistics converge to Poisson-Dirichlet distribution as genus grows.

The three longest components account for approximately 75.8%, 17.1%, and 4.9% of total length.

Results hold for large genus hyperbolic surfaces.

Abstract

We study the length statistics of the components of a random multicurve on a surface of genus $g \geq 2$. For each fixed genus, the existence of such statistics follows from the work of M.~Mirzakhani, F.~Arana-Herrera and M.~Liu. We prove that as the genus $g$ tends to infinity the statistics converge in law to the Poisson--Dirichlet distribution of parameter $\theta=1/2$. In particular, as the genus tends to infinity the mean length of the three longest components converge respectively to $75.8\%$, $17.1\%$ and $4.9\%$ of the total length.