Is random walk drift rational?

For nearest neighbour random walks on certain free products the rate of escape (or, if you wish, drift) was explicitly calculated by Mairesse and Matheus. In particular, their formula (26) gives an example of a "rational" random walk on the free product of $\mathbb Z_2$ and $\mathbb Z_3$ (i.e., essentially, on $SL(2,\mathbb Z)$) with an irrational rate of escape. Yet another (maybe even more explicit) example is provided by formula (28) for the group $\mathbb Z_3*\mathbb Z_3$. I am pretty sure that such examples abound for nearest neighbour random walks on free groups as well - it should be easy to check by using the description of the harmonic measure (which goes back to Dynkin and Malyutov, 1961) and the resulting formula for the rate of escape.