Abstract
Consider an infinite planar graph with uniform polynomial growth of degree \(d > 2\). Many examples of such graphs exhibit similar geometric and spectral properties, and it has been conjectured that this is necessary. We present a family of counterexamples. In particular, we show that for every rational \(d > 2\), there is a planar graph with uniform polynomial growth of degree d on which the random walk is transient, disproving a conjecture of Benjamini (Coarse Geometry and Randomness, Volume 2100 of Lecture Notes in Mathematics. Springer, Cham, 2011). By a well-known theorem of Benjamini and Schramm, such a graph cannot be a unimodular random graph. We also give examples of unimodular random planar graphs of uniform polynomial growth with unexpected properties. For instance, graphs of (almost sure) uniform polynomial growth of every rational degree \(d > 2\) for which the speed exponent of the walk is larger than 1/d, and in which the complements of all balls are connected. This resolves negatively two questions of Benjamini and Papasoglou (Proc Am Math Soc 139(11):4105–4111, 2011).
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Notes
More precisely, for the boundary of a hyperbolic group as above, one can choose a sequence of approximations with this property.
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Acknowledgements
We thank Shayan Oveis Gharan and Austin Stromme for many useful preliminary discussions, Omer Angel and Asaf Nachmias for sharing with us their construction of a graph with asymptotic \((3-\varepsilon )\)-dimensional volume growth on which the random walk has diffusive speed, and Itai Benjamini for emphasizing many of the questions addressed here. We also thank the anonymous referees for very useful comments. This research was partially supported by NSF CCF-1616297 and a Simons Investigator Award.
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Ebrahimnejad, F., Lee, J.R. On planar graphs of uniform polynomial growth. Probab. Theory Relat. Fields 180, 955–984 (2021). https://doi.org/10.1007/s00440-021-01045-5
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DOI: https://doi.org/10.1007/s00440-021-01045-5
Mathematics Subject Classification
- 05C10
- 05B40
- 05C90
- 52C26
- 60D05