Tsviqa Lakrec (HUJI). Affine Random Walks on the Torus

Abstract: I will discuss a quantitative equidistribution result for the random walk on a torus arising from the action of the group of affine transformations. This is a joint work with Weikun He and Elon Lindenstrauss.

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 this 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.

This essentially extends the recently discussed results of Bourgain-Furman-Lindenstrauss-Mozes and He-de Saxcé on quantitative equidistribution of the random walk arising from the action of the automorphism group of the torus.

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