(Question mildly edited for clarity by request of Matt F.)


If $G$ is a finitely presented group, let $|\cdot|$ denote the word metric with respect to a finite set of generators. Suppose $\nu$ is a finitely supported measure on $G$ with rational weights. A random walk on $G$ is built by at each stage multiplying by an element of $G$, selected independently from $\nu$. The *drift* is defined to be the $\nu^{\mathbb N}$-a.s limit of $|g\_1g\_2\ldots g\_n|/n$. 



>  Does there exist a finitely-presented $G$ and a rational $\nu$ so that $d$ is irrational?
> 



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](https://arxiv.org/pdf/math/0509211.pdf). 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.


