On Convergence of Random Walks on Moduli Space

TL;DR

This paper proves the convergence of random walks on the moduli space of Abelian differentials, showing that distributions and paths tend to the affine invariant measure, extending results known for homogeneous spaces.

Contribution

It establishes the convergence of random walks on moduli space in two modes, generalizing known homogeneous space results to this setting.

Findings

Almost every random walk converges to the affine invariant measure.

Distribution of n-step walks approaches the affine invariant measure.

Pathwise equidistribution occurs on the orbit closure.

Abstract

The purpose of this note is to establish convergence of random walks on the moduli space of Abelian differentials on compact Riemann surfaces in two different modes: convergence of the $n$-step distributions from almost every starting point in an affine invariant submanifold towards the associated affine invariant measure, and almost sure pathwise equidistribution towards the affine invariant measure on the $SL_2(\mathbb{R})$-orbit closure of an arbitrary starting point. These are analogues to previous results for random walks on homogeneous spaces.