Large genus asymptotic geometry of random square-tiled surfaces and of random multicurves

TL;DR

This paper investigates the asymptotic geometric properties of random multicurves and square-tiled surfaces on large genus surfaces, revealing probabilistic independence, structural similarities to permutations, and asymptotic formulas for key geometric quantities.

Contribution

It provides new probabilistic and asymptotic results on the geometry of random multicurves and square-tiled surfaces in high genus, including homology independence and cycle count approximations.

Findings

Primitive components are linearly independent with probability 1

Number of components approximates cycle counts of a random permutation

Expected number of components grows as (log(6g-6)+γ)/2 + log 2

Abstract

We study the combinatorial geometry of a random closed multicurve on a surface of large genus and of a random square-tiled surface of large genus. We prove that primitive components of a random multicurve represent linearly independent homology cycles with asymptotic probability 1 and that it is primitive with asymptotic probability $\sqrt{2}/2$. We prove analogous properties for random square-tiled surfaces. In particular, we show that all conical singularities of a random square-tiled surface belong to the same leaf of the horizontal foliation and to the same leaf of the vertical foliation with asymptotic probability 1. We show that the number of components of a random multicurve and the number of maximal horizontal cylinders of a random square-tiled surface of genus $g$ are both very well-approximated by the number of cycles of a random permutation for an explicit non-uniform measure on the symmetric group of $3g-3$ elements. In particular, we prove that the expected value of these quantities is asymptotically equivalent to $(\log(6g-6) + \gamma)/2 + \log 2$. These results are based on our formula for the Masur--Veech volume of the moduli space of holomorphic quadratic differentials combined with deep large genus asymptotic analysis of this formula performed by A.~Aggarwal and with the uniform asymptotic formula for intersection numbers of $\psi$-classes on the Deligne-Mumford compactification of the moduli space of curves proved by A.~Aggarwal.