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Let $\rho$ be a probability measure on $\mathrm{SL}\_d(\mathbb{Z})$ andconsider the random walk defined by $\rho$ on the torus $\mathbb{T}^d =\mathbb{R}^d/\mathbb{Z}^d$. Bourgain, Furmann, Lindenstrauss and Mozes provedthat under an assumption on the group generated by the support of $$\rho$$, therandom walk starting at any irrational point equidistributes in the torus. Inthis article, we study the central limit theorem and the law of the iteratedlogarithm for this walk starting at some point having good diophantineproperties.

Central limit theorem and law of the iterated logarithm for the linear random walk on the torus