First passage percolation on nilpotent Cayley graphs and beyond

TL;DR

This paper extends Pansu's theorem to random metrics on nilpotent Cayley graphs, establishing new variance estimates and analyzing asymptotic cones in first-passage percolation on infinite graphs.

Contribution

It generalizes Pansu's theorem to random metrics with finite exponential moments on nilpotent Cayley graphs and provides variance bounds for certain groups.

Findings

Asymptotic cones are deterministic if and only if the volume growth is subexponential.

Established a sublinear variance estimate for virtually nilpotent groups not isomorphic to Z.

Extended Pansu's theorem to random metrics with i.i.d. edge weights.

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.