TY - JOUR

T1 - Asymptotic shapes for stationary first passage percolation on virtually nilpotent groups

AU - Auffinger, Antonio

AU - Gorski, Christian

N1 - Publisher Copyright:
© 2023, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2023/6

Y1 - 2023/6

N2 - 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-Carathéodory 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-Carathéodory 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.

AB - 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-Carathéodory 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-Carathéodory 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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U2 - 10.1007/s00440-023-01196-7

DO - 10.1007/s00440-023-01196-7

M3 - Article

AN - SCOPUS:85149135248

SN - 0178-8051

VL - 186

SP - 285

EP - 326

JO - Probability Theory and Related Fields

JF - Probability Theory and Related Fields

IS - 1-2

ER -