  We prove the law of large numbers for the drift of random walks on the
two-dimensional lamplighter group, under the assumption that the random walk
has finite $(2+\epsilon)$-moment. This result is in contrast with classical
examples of abelian groups, where the displacement after $n$ steps, normalised
by its mean, does not concentrate, and the limiting distribution of the
normalised $n$-step displacement admits a density whose support is
$[0,\infty)$. We study further examples of groups, some with random walks
satisfying LLN for drift and other examples where such concentration phenomenon
does not hold, and study relation of this property with asymptotic geometry of
groups.
