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Random walks cannot, in general, be pushed forward by quasi-isometries. TameMarkov chains were introduced as a `quasi-isometry invariant' are ageneralization of random walks. In this paper, we construct several examples oftame Markov chains on the free group exhibiting `exotic' behaviour; one, wherethe drift is not well defined and one where the drift is well defined but theCentral Limit Theorem does not hold. We show that this is not a failure of thenotion of tame Markov chain, but rather that any quasi-isometry invarianttheory that generalizes random walks will include examples without well-defineddrift.

Some properties of Markov chains on the free group $\mathbb F_2$