# Recurrence of quadratic differentials for harmonic measure

## TL;DR

This paper studies the recurrence properties of Teichmuller geodesics associated with certain random walks on the mapping class group, revealing typical behaviors related to quadratic differentials and harmonic measure.

## Contribution

It demonstrates that random Teichmuller geodesics are recurrent to the thick part of the principal stratum under specified conditions, linking harmonic measure and quadratic differential dynamics.

## Key findings

- Teichmuller geodesics are recurrent to the principal stratum's thick part.
- Vertical and horizontal foliations lack saddle connections.
- Results connect harmonic measure with quadratic differential recurrence.

## Abstract

We consider random walks on the mapping class group that have finite first moment with respect to the word metric, whose support generates a non-elementary subgroup and contains a pseudo-Anosov map whose invariant Teichmuller geodesic is in the principal stratum of quadratic differentials. We show that a Teichmuller geodesic typical with respect to the harmonic measure for such random walks, is recurrent to the thick part of the principal stratum. As a consequence, the vertical and horizontal foliations of such a random Teichmuller geodesic have no saddle connections.