Affine Random Walks on the Torus

We study quantitative equidistribution of random walks on the torus by affine transformations. Under the assumption that the Zariski closure of the group generated by the linear part acts strongly irreducibly on ${{\mathbb{R}}}^d$ and is either Zariski connected or contains a proximal element, we give quantitative estimates (depending only on the linear part of the random walk) for how fast the random walk equidistributes unless the initial point and the translation part of the affine transformations can be perturbed so that the random walk is trapped in a finite orbit of small cardinality. In particular, we prove that the random walk equidistributes in law to the Haar measure if and only if the random walk is not trapped in a finite orbit.