On convergence of random walks on moduli space

Abstract

The purpose of this paper 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 toward the associated affine invariant measure, and almost sure pathwise equidistribution toward the affine invariant measure on the ${\mathrm{SL}}_2(\mathbb{R})$-orbit closure of an arbitrary starting point. These are analogues to previous results for random walks on homogeneous spaces.