Unimodular random one-ended planar graphs are sofic
Probability
2025-02-14 v2 Combinatorics
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
Cite
@article{arxiv.1910.01307,
title = {Unimodular random one-ended planar graphs are sofic},
author = {Adam Timar},
journal= {arXiv preprint arXiv:1910.01307},
year = {2025}
}
Comments
The extra assumption "one-ended" got added to the title and to the main result, because there was an error in the first version of the manuscript, in the proof of Theorem 3. So Section 2 of version 1 got removed, and other changes were made accordingly. 11 pages, 2 figures