  Our main result is an extension of Pansu's theorem to random metrics, where
the edges of the Cayley are i.i.d. random variable with some finite exponential
moment. Based on a previous work by the second author, the proof relies on
Talagrand's concentration inequality, and on Pansu's theorem. Adapting a
well-known argument for Z^d, we prove a sublinear estimate on the variance for
virtually nilpotent groups which are not virtually isomorphic to Z. We further
discuss the asymptotic cones of first-passage percolation on general infinite
connected graphs: we prove that the asymptotic cones are a.e. deterministic if
and only the volume growth is subexponential.
