Consider an ergodic unimodular random one-ended planar graph of finite expected degree. We prove that it has an isometry-invariant locally finite embedding in the Euclidean plane if and only if it is invariantly amenable. By “locally finite” we mean that any bounded open set intersects finitely many embedded edges. In particular, there exist invariant embeddings in the Euclidean plane for the Uniform Infinite Planar Triangulation and for the critical Augmented Galton-Watson Tree conditioned to survive. Roughly speaking, a unimodular embedding of is one that is jointly unimodular with when viewed as a decoration. We show that has a unimodular embedding in the hyperbolic plane if it is invariantly nonamenable, and it has a unimodular embedding in the Euclidean plane if and only if it is invariantly amenable. Similar claims hold for representations by tilings instead of embeddings.
The second author was supported by the ERC Consolidator Grant 772466 “NOISE” and by Icelandic Research Fund Grant 185233-051.
We are indebted to an anonymous referee for many valuable improvements on the paper, as well as for finding an error and suggesting some ideas for the correction. We thank another referee for the suggestion of the 2nd proof for Theorem 4.2. Thanks to Gábor Pete, Omer Angel, Sebastien Martineau, Romain Tessera and Ron Peled for useful discussions, and to András Stipsicz for a reference.
"Invariant embeddings of unimodular random planar graphs." Electron. J. Probab. 26 1 - 18, 2021. https://doi.org/10.1214/21-EJP665