First passage percolation on nilpotent Cayley graphs and beyond
arXiv:1410.3292
Abstract
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.
The presentation has been improved: some geometric group theory background has been added to make it more friendly to probabilists. The proofs are now self-contained (instead of relying on some other work of the second author)