Almost sure functional central limit theorem for the linear random walk on the torus

Let \(\rho \) be a probability measure on \(\varGamma :=\mathrm {SL}_d(\mathbb {Z})\). Consider the random walk defined by \(\rho \) on the torus \(\mathbb {T}^d= \mathbb {R}^d/\mathbb {Z}^d\): for any \(x\in \mathbb {T}^d\), the walk starting at x is defined by $$\begin{aligned} \left\{ \begin{array}{l} X_0 =x \\ X_{n+1} = g_{n+1} X_n \end{array} \right. \end{aligned}$$ where \((g_n)\in \varGamma ^\mathbb {N}\) is chosen with the law \(\rho ^{\otimes \mathbb {N}}\). Bourgain, Furmann, Lindenstrauss and Mozes proved that under an assumption on the group generated by the support of \(\rho \), the random walk starting at any irrational point equidistributes in the torus. In this article, we study the Functional Central Limit Theorem and the almost sure Functional Central Limit Theorem for this walk starting at some points having good diophantine properties.