Asymptotic shapes for stationary first passage percolation on virtually nilpotent groups
成果类型：
Article
署名作者：
Auffinger, Antonio; Gorski, Christian
署名单位：
Northwestern University
刊物名称：
PROBABILITY THEORY AND RELATED FIELDS
ISSN/ISSBN：
0178-8051
DOI：
10.1007/s00440-023-01196-7
发表日期：
2023
页码：
285-326
关键词：
摘要：
We study first passage percolation (FPP) with stationary edge weights on Cayley graphs of finitely generated virtually nilpotent groups. Previous works of Benjamini and Tessera (Electron J Probab 20:1-20, 2015) and Cantrell and Furman (Groups Geom Dyn 11(4):1307-1345, 2017) show that scaling limits of such FPP are given by Carnot-Caratheodory 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-Caratheodory 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.
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