  Let Gamma be a finitely generated nilpotent group. We consider three closely
related problems: (i) the asymptotic cone for an equivariant ergodic family of
inner metrics on Gamma, generalizing Pansu's theorem; (ii) the limit shapes for
First Passage Percolation for general (not necessarily independent) ergodic
processes on edges of a Cayley graph of Gamma; (iii) a sub-additive ergodic
theorem over a general ergodic Gamma-action. The limiting objects are given in
terms of a Carnot-Caratheodory metric on the graded nilpotent group associated
to the Mal'cev completion of Gamma.
