Hostname: page-component-745bb68f8f-hvd4g Total loading time: 0 Render date: 2025-01-10T23:00:44.787Z Has data issue: false hasContentIssue false
  • English
  • Français

Unimodular random one-ended planar graphs are sofic

Part of: Geometric probability and stochastic geometry Combinatorial probability Graph theory

Published online by Cambridge University Press:  09 June 2023

Ádám Timár
Ádám Timár*
Affiliation:
University of Iceland, Mathematics Department, Reykjavik, Iceland and Alfréd Rényi Institute of Mathematics, Budapest, Hungary
*
Email: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove that if a unimodular random graph is almost surely planar and has finite expected degree, then it has a combinatorial embedding into the plane which is also unimodular. This implies the claim in the title immediately by a theorem of Angel, Hutchcroft, Nachmias and Ray [2]. Our unimodular embedding also implies that all the dichotomy results of [2] about unimodular maps extend in the one-ended case to unimodular random planar graphs.

Keywords

unimodular random graphssoficBenjamini–Schramm limitcombinatorial embedding

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry 60C05: Combinatorial probability
Secondary: 05C10: Planar graphs; geometric and topological aspects of graph theory
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 32 , Issue 6 , November 2023 , pp. 851 - 858
Check for updates
Copyright
© The Author(s), 2023. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Partially supported by the ERC Consolidator Grant 772466 ‘NOISE’, and Icelandic Research Fund, grant number 239736-051.

References

Aldous, D. and Lyons, R. (2007) Processes on unimodular random networks. Electron. J. Probab. 12 1454–1508.CrossRefGoogle Scholar
Angel, O., Hutchcroft, T., Nachmias, A. and GRay (2018) Hyperbolic and parabolic unimodular random maps. Geom. Funct. Anal. 28(4) 879–942.CrossRefGoogle Scholar
Benjamini, I., Lyons, R. and Schramm, O. (2015) Unimodular random trees. Ergod. Theory Dyn. Syst. 35(2) 359–373.CrossRefGoogle Scholar
Benjamini, I. and Timár, Á. (2019) Invariant embeddings of unimodular random planar graphs, arXiv:1910.01614.Google Scholar
Biringer, I. and Raimbault, J. (2017) Ends of unimodular random manifolds. Proc. Amer. Math. Soc. 145(9) 4021–4029.CrossRefGoogle Scholar
Bowen, L. (2003) Periodicity and circle packings of the nyperbolic plane. Geometriae Dedicata 102(1) 213–236.CrossRefGoogle Scholar
Budzinski, T. and Louf, B. (2021) Local limits of uniform triangulations in high genus. Invent. Math. 223(1) 1–47.CrossRefGoogle Scholar
Conley, C. T., Gaboriau, D., Marks, A. S. and Tucker-Drob, R. D. (2021) One-ended spanning subforest and treeability of groups. Preprint.Google Scholar
Curien, N. (2016) Planar stochastic hyperbolic triangulations. Probab. Theory Relat. Fields 165(3) 509–540.CrossRefGoogle Scholar
Droms, C., Servatius, B. and Servatius, H. (1995) The structure of locally finite two-connected graphs. Electron. J. Comb. 2(1) R17.Google Scholar
Elek, G. (2010) On the limit of large girth graph sequences. Combinatorica 30(5) 553–563.CrossRefGoogle Scholar
Elek, G. (2015) Full groups and soficity. Proc. AMS 143(5) 1943–1950.CrossRefGoogle Scholar
Elek, G. and Lippner, G. (2010) Sofic equivalence relations. J. Funct. Anal. 258(5) 1692–1708.CrossRefGoogle Scholar
Hopcroft, J. E. and Tarjan, R. E. (1972) Finding the triconnected components of a graph, Technical Report, Department of Computer Science, Cornell University, Ithaca, New York, 72–140.Google Scholar
Imrich, W. (1975) On Whitney’s theorem on the unique embeddability of 3-connected planar graphs, Recent Advances in Graph Theory (Proceedings of Second Czechoslovak Symposium, 1974, Fiedler, M., Prague, 303–306.Google Scholar
Nachmias, A. (2018) Planar Maps, Random Walks and Circle Packing, Springer Nature, École d’Été de Probabilités de Saint-Flour, 48.Google Scholar
Pestov, V. G. (2008) Hyperlinear and sofic groups: A brief guide. Bull. Symb. Logic 14(4) 449–480.CrossRefGoogle Scholar
Thomassen, C. (1980) Planarity and duality of finite and infinite graphs. J. Comb. Theory Ser. B 29(2) 244–271.CrossRefGoogle Scholar
Timár, Á. (2019) Unimodular random planar graphs are sofic, Preprint, arXiv:1910.01307v1.CrossRefGoogle Scholar
Timár, Á. and Tóth, L. (2021) A full characterization of invariant embeddability of unimodular planar graphs, Preprint, arXiv:2101.12709.Google Scholar
Tutte, W. T. (1966) Connectivity in Graphs. University of Toronto Press.CrossRefGoogle Scholar