Asymptotic shapes for stationary first passage percolation on virtually nilpotent groups

TL;DR

This paper proves that for Cayley graphs of finitely generated virtually nilpotent groups, any Carnot-Carathéodory or conjugation-invariant metric can be realized as a scaling limit of stationary first passage percolation, establishing a converse to previous results.

Contribution

It demonstrates that all such metrics are attainable as scaling limits of FPP, extending the understanding of asymptotic shapes in these groups.

Findings

Any Carnot-Carathéodory metric is a scaling limit of FPP on the graph.

Any conjugation-invariant metric is a scaling limit of FPP.

Conjugation-invariance is necessary for known scaling limits.

Abstract

We study first passage percolation (FPP) with stationary edge weights on Cayley graphs of finitely generated virtually nilpotent groups. Previous works of Benjamini-Tessera and Cantrell-Furman show that scaling limits of such FPP are given by Carnot-Carath\'eodory metrics on the associated graded nilpotent Lie group. We show a converse, i.e. that for any Cayley graph of a finitely generated nilpotent group, any Carnot-Carath\'eodory metric on the associated graded nilpotent Lie group is the scaling limit of some FPP with stationary edge weights on that graph. Moreover, for any Cayley graph of any finitely generated virtually nilpotent group, any conjugation-invariant metric is the scaling limit of some FPP with stationary edge weights on that graph. We also show that the conjugation-invariant condition is also a necessary condition in all cases where scaling limits are known to exist.