Liouville invariance under changing local moves on a Cayley network
Let $\Gamma$ be an infinite finitely generated group with identity $e$, and let $S, T \subset \Gamma \setminus \{e\}$ be finite symmetric generating sets. For $A \in \{S, T\}$, the $A$-walk is simple random walk on the Cayley graph of $(\Gamma, A)$: from $g$ it moves to $ga$ with probability $1/|A|$ for each $a \in A$. The walk is Liouville if every bounded function $f : \Gamma \to \mathbb{R}$ with $$f(g) = \frac{1}{|A|} \sum_{a \in A} f(ga) \qquad \text{for all } g \in \Gamma$$ is constant. Every random walk on a nonamenable group is non-Liouville, and every random walk on a group of polynomial growth whose step distribution generates the group is Liouville. For networks that are not transitive, T. Lyons showed that the Liouville property can change under a change of conductances by a bounded factor. Erschler and Frisch proved that the $S$-walk and the $T$-walk are Liouville or not together when $\Gamma$ is a finitely generated linear group of positive characteristic.
Problem. For every choice of $(\Gamma, S, T)$, is the $S$-walk Liouville if and only if the $T$-walk is Liouville?
Source: Kaimanovich and Vershik (1983); see Lyons and Peres (2016), Section 14.7, the notes after Theorem 14.49.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- PR — Probability
- Secondary
- GR
- Keywords
- AIM Open Applied Problems, Liouville property, Poisson boundaries, random walks on groups
References
@article{kaimanovich1983,
author = {Kaimanovich, V. A. and Vershik, A. M.},
title = {Random walks on discrete groups: boundary and entropy},
journal = {Ann. Probab.},
volume = {11},
number = {3},
year = {1983},
pages = {457--490},
doi = {10.1214/aop/1176993497},
}
@book{lyons2016,
author = {Lyons, Russell and Peres, Yuval},
title = {Probability on Trees and Networks},
series = {Cambridge Series in Statistical and Probabilistic Mathematics},
publisher = {Cambridge University Press},
address = {New York},
year = {2016},
doi = {10.1017/9781316672815},
}
@article{lyons1987,
author = {Lyons, Terry},
title = {Instability of the {L}iouville property for quasi-isometric {R}iemannian manifolds and reversible {M}arkov chains},
journal = {J. Differential Geom.},
volume = {26},
number = {1},
year = {1987},
pages = {33--66},
doi = {10.4310/jdg/1214441175},
}
@misc{erschler2022,
author = {Erschler, Anna and Frisch, Joshua},
title = {Poisson boundary of group extensions},
year = {2022},
eprint = {2206.11111},
archivePrefix = {arXiv},
}
@misc{aim-liouville-generating-set-invariance,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Liouville invariance under changing local moves on a Cayley network}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/184-liouville-generating-set-invariance.md},
note = {Entry 184 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00450, "Liouville invariance under changing local moves on a Cayley network", clopen.solutions.