Affine random walks on the torus
Weikun He, Tsviqa Lakrec, Elon Lindenstrauss
TL;DR
Addresses affine random walks on $T^d$ and provides quantitative equidistribution estimates depending only on the linear part, under the assumption that the Zariski closure of the generated group acts strongly irreducibly and is either Zariski connected or proximal. Establishes a dichotomy: for every starting point x, either $\mu^{*n} * \delta_x$ converges to the normalized Haar measure on $T^d$ or the $H$-orbit of x is finite (i.e., the walk is trapped in a finite set), under these irreducibility/proximal or connected assumptions. The analysis focuses on finitely supported $\mu$ on SL_d(Z) ⋉ T^d, with $H$ the group generated by the support and $\Gamma$ its projection to SL_d(Z), satisfying the same hypotheses. The results extend the quantitative linear equidistribution theorems of Bourgain–Furman–Mozes–Benoist–de Saxcé (BFLM) and de Saxcé–Hauser (HS) to affine actions, clarifying finite-orbit obstructions and recovering Boyer’s Diophantine special case as a related instance.
Abstract
We consider random walks on the torus arising from the action of the group of affine transformations. We give a quantitative equidistribution result for this random walk 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. Specifically, 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.