First passage percolation on nilpotent Cayley graphs

We prove an asymptotic shape theorem for the standard first-passage percolation on Cayley graphs of virtually nilpotent groups. By a theorem of Pansu, the asymptotic cone of a finitely generated nilpotent group is isometric to a simply connected nilpotent Lie group equipped with some left-invariant Carnot-Caratheodory metric. 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 the companion work[24], the proof relies on Talagrand’s concentration inequality, together with Pansu’s theorem. Adapting an argument from[3] 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.