Open Access
2022 Random walks on finite nilpotent groups driven by long-jump measures
Laurent Saloff-Coste, Yuwen Wang
Author Affiliations +
Electron. J. Probab. 27: 1-31 (2022). DOI: 10.1214/22-EJP745


We consider a variant of simple random walk on a finite group. At each step, we choose an element, s, from a set of generators (“directions”) uniformly, and an integer, j, from a power law distribution (“speed”) associated with the chosen direction, and move from the current position, g, to gsj. We show that if the finite group is nilpotent, the time it takes this walk to reach its uniform equilibrium is of the same order of magnitude as the diameter of a suitable pseudo-metric on the group, which is attached to the generators and speeds. Additionally, we give sharp bounds on the 2-distance between the distribution of the position of the walker and the stationary distribution, and compute the relevant diameter for some examples.

Funding Statement

Y.W. is partially supported by NSF grant DMS-1645643 and Austrian Science Fund (FWF) project 34129. In addition, Y.W. and L.S-C. are both partially supported by NSF grant DMS-1707589.


We thank the anonymous reviewer for carefully reading our manuscript.


Download Citation

Laurent Saloff-Coste. Yuwen Wang. "Random walks on finite nilpotent groups driven by long-jump measures." Electron. J. Probab. 27 1 - 31, 2022.


Received: 22 July 2021; Accepted: 25 January 2022; Published: 2022
First available in Project Euclid: 15 February 2022

MathSciNet: MR4381058
zbMATH: 1486.60006
Digital Object Identifier: 10.1214/22-EJP745

Primary: 60B15

Keywords: group , mixing time , Random walk

Vol.27 • 2022
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