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Unimodular random one-ended planar graphs are sofic

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

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: madaramit@gmail.com
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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

Information

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 32 , Issue 6 , November 2023 , pp. 851 - 858
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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Footnotes

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

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